Poseidon really does treat Percy like a kitten a lot. Currently Percy has fallen asleep in his dad's arms and it doesn't look like Poseidon will be waking him up any time soon.
But on the following days on Olympus (the demigods are allowed to stay for longer), they see Poseidon scruffing Percy and lifting him up, causing Percy to just go limp. And also they've seen him frequently play with Percy's hair which seems to relax the demigod and settle his ADHD restlessness so much. They also see Percy be carried around or cuddled whether he wants to be or not.
Percy seriously abuses his power over the sea god to get what he wants (mostly for just small things). Poseidon is already a sucker for baby seal eyes from Percy so when Percy is being all sweet and calls him papa at the same time he gets such good cuddles out of it.
And Percy was a manipulative little shit who knew far too well how much power he had over his overprotective papa and he had no qualms about abusing said power. One memorable incident being when he was bickering with his godly cousins (Hermes and Apollo) over something minor and turned from the aggravating bitch he was being to his cousins, to smol baby mode when he spotted his dad, knowing his dad could scare his cousins easily enough.