Keep in mind I am on season 10 of Supernatural, and if anything I say is wrong, please take it in stride and with kindness. Thank you!
The song "Blood stream" by Alyssa Grace fits Gabriel beautifully and I'm here to dissect why through lyrics and long paragraphs!
"You gotta love all that you have and I'll take that to my grave
Cause soon you'll open up your eyes and it'll all be gone someday"
There are two ways to look at this for Gabriel, starting with the most obvious being his family. Gabriel loves his family, he says so in Changing Channels and when he refuses to fight Lucifer. He cares about them, regardless of their actions against others, and they were presumably close in Heaven before all of creation. One day, of course, Lucifer decided to betray God, got into fights, starting causing shit in Heaven, so he got booted down to Hell, and then at some point God left. Which means, in a portion of time hard to desipher, Gabriel lost his brother, and his father, with all of Heaven in unknowing bliss. He literally lost everything he had. However, the second way to look at it applies to humanity.
Gabriel loves humans. He chooses them over Lucifer, sides with them over Micheal, tries to help, however minimally or majorly, in stopping the apocalypse. However, despite that fact, for the longest time he didn't, even when he did he didn't put all his eggs in that basket. That's because he truly believes in the apocalypse. Despite him making a family with the Pagan Gods and having a kinship with monsters because he also kills people who are unmoral, with humanity, he still chooses to believe they will all die. He loves them anyways, even though they will be 'gone someday', the lyric is how Gabriel approaches mortals and everything around him, that he knows will all be gone one day thanks to Micheal and Lucifers fighting. He literally takes it to one of his graves when he dies for them, despite the fact he'll outlive it all, or would've.
"And I'm sitting here convincing you to pick up the phone and call"
Okay, this is entirely me speculating, but Gabriel is the archangel of delivering messages. Meaning other than Joshua, he was likely speaking with God a lot to deliver what he was saying to the angels below who never even saw God. He very well could have been trying to get God to talk to the other angels more, because apparently, according to Metatron, he hated sending information to humans and other angels. So he easily could have been telling God to just talk to humans, or at least spread basic information. This is just a headcanon of types of course, I just think the connection between the lyric talking about communicating and Gabriel being the angel of communicating was an interesting comparison.
"I am my fathers daughter but I'm nothing like him at all"
Gabriel is literally Gods creation, and despite this, he loves humans. It can be assumed, from the fact that God left, that he doesn't really care what happens. According to Joshua, he considers it, 'not his problem' and while I will return to this idea because it SEEMS a lot like Gabriel with another lyric, Gabriel actually doesn't believe this. Sure, he kills people that mostly deserve it as a trickster to play the part, but to be realistic, he doesn't HAVE to. Despite this, unlike God, Gabriel does care, he does eventually step in to save Sam and Dean, to make sure everything goes right, sure, but still, he makes the move. He steps in to teach Sam a lesson, to try and teach him to learn and live without Dean, to teach him this fixation isn't healthy. He tries to make the boys learn to play their roles, tries to talk Lucifer down from hurting people he actually cares about, and thats the important part. Gabriel cares. When he doesn't have to step in, he still does, and God doesn't. No matter what, even when his creations beg him to, he doesn't care. Gabriel loves every bit of humanity, and his family, and those he wants to protect. He loves too much, and that's why he runs. Not because he doesn't care. He is nothing like God in that regard, which is a fundamental difference in their shared trait.
"I hear him in the way I speak
Or when I look in the mirror"
Gabriel was around for all of creation. While the pagan Gods confuse me, and where they came from confuses me, because GOD God is real, and so are they his creations, is Loki Gods creation, it truly doesn't matter because Gabriel isn't Loki. He sees things in multiple dimensions, including his wings, and his true self when he looks in a mirror. He, who literally sees his dad every time he looks in the mirror because of the human form, something he watched God make, as an archangel, Gods first-ish creation, but also his grace. Whenever Gabriel speaks or moves his grace, something we know makes sound, he hears his father in it, the man's creation, the magic God engineered only because only God can make an angel. He hears and sees his Dad in everything he does, because he literally is built from him, every molecule. The one thing Gabriel can't run from. I also think, back to the archangel of delivering messages, when he speaks from what God wrote, he could hear him in the back of his head when he spoke God's message. (Honorable mention to a hilarious post I saw somewhere of Gabriel delivering the message to Mary that she's pregnant with God's child. Hilarious.)
"That I'm like him cause he made me the way I am when we're fighting"
Back to the serious stuff, this is what Gabriel and God have in common. Gabriel runs. He runs whenever people acknowledge the connection he has to things, to the Gods, to humans, to his siblings, and how much it hurts. He ran from their fighting, and God left because he just gave up. While they leave for two different reasons, it doesn't change the fact that when fighting comes, Gabriel is the same as his father, and runs. And God made him that way, it's something they share, that, 'It isn't my problem' mindset. Only for Gabriel, its an act and coping mechanism, for God, He's just like that.
"And I'm craving for you to change for me"
Okay, bit of speculation, bit of truth, but Gabriel obviously had to deal with the fighting in Heaven, but, he asked Lucifer to stop. He wants them to stop, to join his side, because he sees a way out, at least at the end, to escape the plan, and he wants his siblings to come with him. They all feel this way, it seems, with Lucifer trying to convince Micheal to damn it all, but Micheal is too stuck on Gods plan, and Lucifer is too busy throwing a tantrum to actually leave like Gabriel wants him to, and like he offers Micheal. But Gabriel doesn't want to fight, and they all want the others to change so they can become a family again, and that includes Gabriel. Such is the misery of the angelic family.
"Wondering what it takes for you to stay
Off my mind and far out of my bloodstream"
This is the entirety of Gabriel trying to escape through Loki. He throws himself at humanity, and having fun, and becoming a trickster, indulging in everything. Women, candy, the ways of fun, clear distractions. He wants to avoid his family, his blood stream, and he doesn't want them to find him. Gabriel for the majority of the Earths creation was just running around, laying low, and avoiding his family, avoiding them being in his bloodstream, he literally makes a deal to share Lokis identity, aka, getting rid of his connection to the angels. He makes new family in the Pagan Gods, and this lyric is definitely the one that made me connect the song to him, because it is the definition of Gabriel's shtick.
"But I should've known that you'd gone mad
When I saw the way you'd punch your doors"
Okay, again, pure headcanoning family slop here, but in Hammer of Gods, right before Lucifer kills Gabriel, he mentions he taught Gabriel all the tricks he learned, implying they spoke a lot in Heaven, and almost that he and Lucifer were close. Considering he loves the man, and they were brothers, plus spending time together to learn and commit tricks and mischief, Gabriel probably had to watch Lucifer, as he went from Gods favorite, to humans coming. Lucifer could very well have likely tried to convince Gabriel to help him give humans the apple, framing it as, 'another trick ' on God, only for Gabriel to deny because he knew it was too far. Regardless, this lyric could aid to the thought of Gabriel, who should've known his brother would fall after watching him slowly get more jealous and angrier in his defiance with God, till he was cast out of Heaven.
Also, I haven't gotten to season 13, and I've been running from SPECFICALLY Gabriel related spoilers, because spoiler alert, he's alive, but I got a snippit, but I feel like the lyric "Something deep inside of me tells me that your not better now" is gonna relate between Lucifer and Gabriel, but we shall see.
"Why don't you want this anymore"
Again, speculation into the mans mindset, but there HAS to be some frustration from his part, because of all of his families fighting, them splitting up, and God's leaving. It must've felt like he was the only one who wanted to stick together, because everyone else just wouldn't stop fighting. Why is he the only one who cared enough to say 'damn the plan' and run away, or just not pick a side?
Fun little after note, which is more of a wish, I wish they made it seem like Raphael loved his family as much as the others did. They kinda don't mention that he's the others brother at all, and his motivation doesn't connect either. I mean, I would've loved any mention, y'know? Like, maybe a plot twist where he wants to break both his brothers out of the Box because he wants them free, or he just wants Micheal back and will risk Lucifer getting out because he spent the most time with Micheal and misses him. Maybe it's cause he wants the apocolypse to still go down because Micheal wanted it more then anything and he wants to make him proud. He wants the status quo because something- SOMETHING! Everyone else gets a brother scene, hell, Lucifer even gets mad when Castiel firebombs Micheal, knowing very well it only hurt him and didn't kill him. Also why do they never use that anymore? Anyways, it feels like Raphael is the only one with no connection to his brothers, and then he dies. Makes me sad, most people forget him in fanfiction too. Love for the forgotten Archangel brother.
Hope you guys enjoyed my 1,800 WORDS WORTH of ramblings! Keeping in mind, I had to rewrite this entire thing, oh boy, I can't believe I spent an hour and a half on a Tumblr post.