Toho is no stranger to retroactively inserting films into the continuity of a newer film, or even having films take place in multiple continuities. For example, Mothra exists simultaneously in the continuity of the Showa Godzilla series and the Kiryu Saga, both of them retroactively since the filmmakers in all likelihood were not concerned with the film fitting into the continuity of another one. (I think Japanese Wikipedia says that the original script for Mothra contained an allusion to Godzilla, but I don't know how true that is.)
It's much the same thing with how G98 is handled in the context of GMK. The 1998 film was created as a standalone film with no continuity with other movies, but for GMK Toho just disregarded that and placed it into the continuity of that film, by extension placing it after the 1954 original. The book Godzilla Dictionary, re-published as recently as 2014, lays out an extensive timeline for the series and says that the 1998 film is in continuity with GMK, and the "monster similar to Godzilla" is in fact the monster from that movie. I think it's feasible in the universe GMK establishes. Few eyewitnesses or physical evidence of Godzilla's original attack remain, and many details were covered up by the government. Godzilla hasn't been seen since then, and humans with their tendency to deny tragedies end up writing Godzilla off as a legend. Thus, I could see a giant monster that suddenly appears in the U.S. to get identified as "Godzilla" with a majority of the public not having any recollection of the monster existing. GMK shows us this as well. When Baragon appears he is mistakenly called "Godzilla," and characters are shown to regard Godzilla as a myth and joke about his attack as if it never happened. I think it fits Toho's track record for a slightly modified version of the 1998 film to exist in a continuity with G54 and GMK, while simultaneously occupying its original intended continuity where it is standalone and shares no ties to the other movies, much like how the 1954 movie simultaneously exists in countless different continuities, sometimes in modified form.
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