On the topic of this Godzilla's identity, while the 1999 guide book is imprecise about whether it is the 1954 Godzilla or a second one, I think the fact it calls it a "fourth generation of Godzilla" suggests that even within its continuity it is not the same individual as the 1954 Godzilla. That book along with
Shin Godzilla Walker also demonstrate that
G2K is considered to follow the original film, even though no such dialogue is present in the film itself.
Most Japanese sources I've seen, barring terrible mistranslation on my part, seem to suggest this Godzilla is indeed meant to be a second Godzilla that appeared after the 1954 Godzilla. Godzilla.jp, Toho's former official site for the Godzilla series, said that it was a second Godzilla that appeared after the first Godzilla (if anyone could find an archive of that site's page for
Godzilla 2000, we could find the actual text), and
Japanese Wikipedia's bio for this Godzilla opens with: "1954年に出現した初代ゴジラの後に日本にたびたび上陸している緑色の体色のゴジラ。" To the best of my understanding, it says something to the effect of "After the first generation Godzilla appeared in 1954, this green Godzilla has frequently landed in Japan."