Think of the reason you're giving names to notes: communication.
So, it depends who you're communicating with, their expectations of you, and your expectations of them.
When I play soprano ukulele, I think in terms of the guitar fretboard - the intervals between strings are the same as the top four strings of a guitar. So in my head, I play a guitar "D" shape, and my head thinks "D". But in fact, that's a G.
If I'm playing solo, that does no harm, but I'm duetting with (say) a pianist, though, there will be problems from the start. I say "OK, we open with a D major chord", I strum my D shape, which is actually a G. The pianist plays D, F#, A, and it sounds terrible.
The same happens if you're playing with a bassist who's tuned conventionally, while you're downtuned. You shout "Louie Louie in G", you strum a G shape, which is a Gb chord. The bassist plays an actual G, and it sounds awful.
Of course if the whole band tunes up or down the same amount, then it doesn't matter.
Otherwise, you need to agree how to communicate.
- The easiest thing for your fellow musicians, is for you to communicate using the real pitches. That way, they don't need to know how you've tuned your instrument.
- However, if they're better at mental transposition than you are, it might be OK to shift that load onto them. "OK, guys, I'm playing with a capo on the third fret here, so when I say 'A' I really mean 'C', is that OK?"
- Or you could dodge the issue by talking in relative intervals (after agreeing a key). Instead of "C, F, G", "One, Four, Five".
Orchestras have a similar problem: quite a few brass and woodwind instruments are "transposing instruments", in which the note names don't match standard pitches. Scores written for these instruments are transposed accordingly, and the conductor must remember to use the note names expected by those players, when speaking to them.