Summer truly is a more well-adjusted Beth, and I think Beth envies her a lot for that in ways that are noticeable during episodes.
It doesn't mean Summer won't experience adolescent troubles like similar social discomfort, rejection, confusion, etc. - we know that she does, because that S8 ep shows us she's excited to go through adolescence again, but with the 'advantageous' mindset of an adult. Shit, wouldn't most of us be?
These moments are where Beth can be a shoulder for Summer, and we frequently see them bond over the more colloquially 'feminine' world of social connections and gossip. However, there's a world of 'normal' familial dynamics that Beth has disconnected herself from through her substance use and her bitterness. It's hard for her to fall into a normal family life after her incredibly dysfunctional childhood. Even keeping her last name feels indicative of their disconnection in the family. It's not that changing her last name would promote cohesion - no one should change their last name because marriage unless they want to.
It's more that when Rick disappeared, a last name was all Beth had to remember him by, and I don't think she'd change it for the world - even if it didn't have extra meaning behind it. It's that all she has is a word to scribble down on legal documents that burn when she scrawls them across pages. It's a piece of her that she holds onto, despite the rage and the hurt, because she wants to pull it back into her life.
In a lot of ways, I think Beth resents Summer (and Morty, to a lesser degree) for what Beth has been able to provide for her children through the sacrifices that she's made, which weren't made for her. Between resenting her children for what she couldn't have, loathing her husband as the catalyst for her situation, and mourning/hating her father for abandoning her... It's not particularly surprising Beth turned to the (genetically predispositioned) path of alcoholism to avoid being an active parent in her family.
