Do you think it's rude when a family member makes you compete with another family member?

yes i do think it is rude espesscially over something es stupid as that

Well, your aunt probably feels as if she didn’t do as well of a Job at raising her daughter as your mom did you, so she may think that she can somehow make her daughter surpass your level quickly thus achieving the level of content or self accomplishment as a great parent that she would have liked to be, which might not even be able to blame her since you say she’s bipolar, but to answer your question, yes it is totally rude, I know people and some “family members” like that, me personally, I just don’t associate whith them, and when they’re saying something smug i strike a conversation with someone else or simply walk away.
I really hope this helps. Simply Something

This is your aunt’s way of competing with her sister (your mom), through you and your cousin. You don’t have to let it become something between you and Audrey. If you don’t go into it with a competitive attitude, or caring about who is better, maybe you can turn it into a fun family evening of making music together. Give Audrey some praise for her performances; encourage her. If you need to, talk to her about the competition between your moms when they’re not around, and tell her you don’t want it to poison your family relationship with her.