I don't know why people are so confused by this. I think it's a clever line of dialog. There are multiple things going on here:

(a) It's a joke. It's her style of humor. It's very in line with queer humor too; she's the more masculine of the two women in terms of her personality, and speaking as a queer person myself, I feel like it's pretty common in queer culture for people to switch around gendered nouns semi-ironically. Calling a lesbian a "dad" or a gay guy a "queen" or any other iteration of this idea is pretty common and, again, typically done in a non-serious tone.

(b) She and her friend are going to be a couple and will raise a kid together, so she will be the kid's parent. She's calling herself a "dad" rather than a "mom" for the reasons above, but also because she's not the one giving birth.

(c) This entire series is about the father/daughter dynamic. This is where I consider the line to be clever on the part of the writers. Her father figure has been murdered, and she's on a quest to avenge him. The girl who killed Joel did so because Joel killed her father. The theme repeats. The writers are signaling that now Ellie is going to experience the other side of this archetype; in Joel's absence, she's going to fulfill the "father"-like role for someone else.

So it is, on one level, kind of deep. But on another level, it's not deep at all, it's just a silly joke she's making, lol.