Kaiju no Mura, I think the misunderstanding that you feel the names a company uses for the product (in this case, a film series) have to 100% accurately reflect established facts. And the truth is they don't... a company can call they own property whatever they want, accurate or not.
Here's one obvious example: JURASSIC PARK. The books and the movies features prehistoric animals from all three periods of the Mesozoic era, and the most of the key dinosaurs (Tyrannosaurus rex, Velociraptors, Triceratops, etc) are from the Cretaceous, not the Jurassic. Crichton went with "Jurassic" because he liked the sound of it, and his publisher, Universal studios, Steven Spielberg all followed suit.
Now with three Godzilla films series, Toho refers to the 1954-1975 movies as the "Showa Godzilla Series", the 1984-1995 movies as the "Heisei Godzilla Series", and the 1999-2004 films as the "Millennium Godzilla Series". I know this because I've worked with/for Toho many times over the past ten or so years and I am friends with a number of Toho filmmakers and execs and they use these terms when discussing the films. And they've been doing so for a long time.
The seven movies from 1984-1995 movies are all linked together by story continuity, so to Toho they are one series. With six of the seven films released during the Heisei period, calling these films the "Heisei Series" is a very simple way to immediately differentiate them from the first fifteen Godzillas which were all produced during the Showa period. Sure Toho could have called them the "One Showa and a Bunch of Heisei Godzilla Series", but like JURASSIC PARK sometimes simpler is the way to go. Toho also refers to the second series as the "VS Series", but there's no "VS" in the title of the 1984 film so Toho is obviously not worried about the names they use being 100% accurate.
Toho went with "Millennium Series" for the 1999-2004 films to-- again-- create a clear distinction from the previous series. Before realizing that Toho had already named the new series some American fans coined the term "Shinsei" for these films. I've never cared for fan names... I know Toho names everything so I go with what the people who make and own these films say rather than what some fan who had nothing to do with the productions thinks up. If Toho didn't use the terms Showa, Heisei, Millennium for their Godzilla series I wouldn't either. But they do.