Add it if you want, don’t if it’s not a part of your vision. That way it doesn’t come off as forced, or disingenuous. You should never allow yourself to be pressured into doing something just because your audience wants it. If it doesn’t stick, it’s not meant. Authenticity will keep you more engaged in your projects.
I will say this though - just to give you an overall clear view of the bigger picture - labels are not inherently bad. They do serve their purpose from time to time. Without labels, we wouldn’t know what was dangerous and what wasn’t. Labels can be helpful when we are trying to gather information. i.e.: If you’re someone just coming to realize your sexuality, and/or gender, then labels can be something that help you as you discover your identity; something that helps you define how you go about your life. They are also a great way to create and find communities. On the flip side, this can also be used for people to pigeonhole someone to a specific label - as if they can’t be interchangeable. 
This is when I believe it becomes an issue.
Depending on the person, this viewpoint obviously changes. Some people don’t like any kind of labels - like your friend. In history, labels were created to separate us from them (this is me speaking as a Black womxn). It can leave people feeling ostracized, or left feeling as other. I myself, actually like labels. If you know of them, are aware enough of when/how to use them, you can make informed decisions regarding certain situations…
I say all of this because for some people, knowing where certain people stand can be beneficial… We live in a world full of diversity where no two people are the same, so if one person in your fic isn’t willing to identify as one thing or another, then that’s absolutely fine. Just remember that it also comes at a price. People reading your work have chosen to disassociate themselves from real life. If they can’t find/see themselves in the world you’ve created, they may have trouble connecting to the story. Even possibly abandon it all together.
And regarding the Phineas and Ferb reference, I don’t personally agree that’s the right way to go about it. It’s very much giving me that comment Transphobic Rowling essentially made when she said that Hermione was NOT written with a specific race in mind. There are so many reasons why that is WRONG, but the main point is that when refusing to acknowledge the differences of people, you are ignoring who they are. When Rowling said that, I assume she thought that would appease us, but what that did was confirm what the Black Community already knew when it came to the Harry Potter Universe, and it’s that diversity was not Rowling’s goal. It also showed she thought so little of us to think we would be fine with scraps. That instead of just stating she had no issues with Noma Dumezweni, who happens to be a black woman, casted as the roll of Hermione Granger she went straight to pandering to black women. While race doesn’t need to be thrown in readers faces, there is something to be said when someone is willing to ignore the character who they have obviously written one way only to change it for ulterior motives - but I digress… Sometimes, by not making clear your intentions, it could cause more harm than good… It reminds me of this one line from Hamilton when Alexander asks Burr, “if you stand for nothing, Burr what will you fall for?” It’s a more extreme circumstance sure, but something to think about all the same.
There’s certainly a duality to the subject. Since this is a story, it may be a bit easier to use this as a topic of discussion/debate with open dialogue about how some of the characters feel on the topic… Or, you can continue the “Show don’t Tell” route it seems you’re on. There’s no 
real answer that you can get from us. This has to be something you decide for yourself.
Good luck. 