NCC-7100 is labeled with a name. The name is on the lower nacelles. It is identified as the U.S.S. Stargazer.
Do you have a pic as source? In every single photo I've seen of that desktop model, I have not seen the name "Stargazer" printed on it, just the 7100 registry. There is a pennant on the lower nacelle, but the writing is too small to make out any words (unless you have a better pic than this one):
(And if that decal was taken from a TMP Enterprise kit like the rest of the model, then that decal should read "Starship U.S.S. Enterprise United Federation of Planets.")
The Enterprise was constructed third in the Galaxy class series, the Galaxy first, Yamato second. Technically the Enterprise is the junior ship.
What does that have to do with how long it takes to build a Galaxy class ship? Blssdwlf theorized that the Yamato in "Contagion" was a newer, different ship built after the Yamato seen in "Where Silence Has Lease." What does the age of the Enterprise have to do with that?