Hmh? We hear her called by at least three class names over the course of her onscreen service: "Starship class", "Constitution class" and "Enterprise class".
It's the very same ship in all three cases. Or four, in fact, because "Constitution class" applied to her both back in the days of DSC and in the days of ST6:TUC, while "Starship class" was there during TOS and "Enterprise class" was used in ST2:TWoK, both in between the "Constitution class" periods.
It's not particularly objectionable, really. Every time the class name changes, the configuration of the ship also changes, and indeed often quite drastically so! In contrast, the slight variation in nacelles, dishes and bridge shapes during TOS is not known to have altered the class name.
Where the class name in each particular case comes from is a separate issue. ITRW, it would come from the first ship brought to this configuration, so that, say, a major modifying of USS Enterprise of Constitution class to a new configuration, or a new engine standard, or a new weapons outfit or whatever, would make her the spearhead of the all-new Enterprise class, and others likewise refitted would then belong to Enterprise class. Starfleet practices don't need to differ from this, in light of the evidence. (We may or may not wish to postulate the existence of a USS Starship there...)
Timo Saloniemi