Precisely. So with an A chord shapr one fret up kt becomes Bb, two up it becomes C, then C#, then D and so forth up to 12th fret (if you could get that far), whenit becomes A again.
The only time it matters what you call it is if you're playing with other people, and they change key and shout out "E" or something, and you've got to work out what shape that means for the place where you are capoed.
Capos aren't just there to shift the sound to match your voice. There are sounds you can get with a capoed guitar that aren't really possible without, even if you're really good with making with barre chords.
For example, playing in the key of D but using G chords and a capo on the 7th fret is a very different sound and feeling. When two guitars are playing together, it can be a good idea to have one capoed high playing different chords.