What does Arya want to be?

I don't think she's thought it out. She wants to re-establish House Stark and to be a badass, now that both wishes have come true she's just enjoying them rather than thinking about what she'll do with the rest of her life. And she won't think it out in the near future either, not when the news about the Night's King crossing the Wall hits. She'll go to war with Jon and Dany's dragons, and thank the Seven that she's not back at Winterfell trying to feed the refugees and deal with the bannermen's egos and nurse the wounded and trying to counter Cersei's moves and fending off any besieging armies that happen by. That is what Ladies are supposed to do in wartime, in medieval societies, and that's what Sansa will be stuck with doing.

Who knows what her future will be, or whether she has a future all. You don't take power from Death Gods and expect to live a long and untroubled life afterwards.

Men and women have different roles. The role of a Lady is very narrow and the status is below men. Basically, do what one's husband or father tells her. He dictates who she will marry. Cersei didn't want to marry King Robert nor Loras. Daenerys didn't want to marry Drogo. Ditto Myrcella.

Look pretty, marry well, play music, do needlework and learn neat cursive writing, run a household. Not surprising since that's historically been the norm in the real world, too.

Her opinion isn't worth much. Notice how Jon Snow didn't take Sansa's advice seriously nor Robb with his mother re: Fry. A true lady isn't a leader, but supports the men and raises the children. Arya is fighting against being pigeonholed which is why she rejects that term.

Arya clearly enjoys fighting which is a male endeavor. That's the reason why her father called her fencing lessons "dancing" which is acceptable for a "lady".

In Westeros, women tend to become warriors or leaders when the male heirs have all been killed off like with Brienne, Yara (can't count Reek), Dani and Lady Mormont. Sansa is still below Jon Snow in status and only acting leader until he returns even though he's illegitimate and has no legal claim.

I think she started out wanting to be something like Brianne, but later events lead her to become what she is and she's rather enjoying it.

No one.

Maybe she keeps saying she doesn't want to be a lady because she honestly doesn't want to be one. She wants to live like a dude. Not as a dude, but to live freely like one with a responsibility only to herself, like how Braun lives.

Arya seems to want to live her own life, on her own terms. She doesn't want to take shit from anyone. She doesn't want a husband telling her what she can and cannot do. Or I think, the responsibility of raising children.

I found it interesting that despite Jaquen saying the faces were meant for no one, and are like poison if you haven't renounced your identity, Arya continues to use the power of the Faceless Men with no ill effects. You might have expected them to come after her. After all she's taking their secrets out into the world and using them for her own purposes. What if she decides to train others? But, nothing. She hasn't exactly been keeping a low profile yet the Faceless Men have made no effort to rein her in. I can't help but think there will be consequences for Arya though. Some price to pay; maybe her own life, or maybe it will be more complicated, but gods of myth never grant mortals power without demanding something in return.