Ultimately, your musicianship takes priority over the instrument. In that sense, the instrument is not so much of a concern, and certainly not the name on it. If that is your question, then no, not at all.
I will say that I play a bit differently on different instruments. But, there again, the name has zero to do with my interaction with the specific instrument; more the varying characteristics of the instrument.
Now, when you get to auditions or the studio - whole different story. And, as I mentioned above, since a player tends to interact a little differently with each instrument, one needs to be aware of those differences and try to fit the whole thing to the specific gig. In some cases, yes most definitely, the name on the headstock can determine whether the band will hire you or the sound guy will work constructively with you. Maybe more subtly, whether the band or some members of the band are comfortable playing with you, so you can focus on the music together. Making music is usually not a solo effort. One needs to be aware of all the surrounding interactions that are going on, musically and personally.
Don't know if that helps, but IME.