Yes, the guitar you use will have an impact on the final result, but more in terms of how close it gets to the original amp and how quickly than in making the profile sound like that guitar through that amp.
It's limiting down the frequency range and dynamics by the instrument and it's pickups basically, if you use a guitar with very loud/compressed pickups that are quite dark sounding then your profile may not sparkle as much as if you'd used e.g. a Telecaster
nor have the accuracy to the original amp in the treble range and dynamics and transients, simply because you haven't given it that information. Conversely refining with a Tele may result in your profile being less accurate in the bass range. TBH the end result wouldn't be dramatically different, but playing with the other guitar against the original you may notice some discrepancies.
I find refining with a superstrat means you have a pretty wide range of tones (HB, SC, Piezo) so I don't get many surprises when I use a different guitar.