Like the previous few films, this movie ignores all other entries in the franchise after the 1954 original. Unlike them, however, it does acknowledge other Toho films.
Because this is a scene that needed re-showing.
The government's solution to this made obvious by the film's title: they build a robot Godzilla to fight him (much as they did in the similarly-titled 1993 film).
Of course, as a plan of action it's even crazier than it is obvious, because they build this new Mechagodzilla over the skeleton of the original 1954 monster, and use cell samples from those remains to create "DNA computers" that will control the machine. About the only thing they don't do to ensure disaster is have someone say "nothing can possibly go wrong!".
I suspect this film may have had a formative influence on Guillermo del Toro as the scenes where Mechagodzilla is transported remind me of sequences from his recent kaiju vs giant robot effort, Pacific Rim. Unfortunately, it's not nearly as entertaining as del Toro's film.
I mean, Godzilla Against Mechagodzilla has got some decent enough giant monster action ... it's just that it seems to lack commitment. At the very least, it sets stuff up but doesn't use it well. For instance, the "DNA computers" in Mechagodzilla go wild after its first encounter with Big G, causing it to go on a rampage ... but then it runs out of power and in the next scene, the problem has been fixed. Or there's the pilot of the robot, who froze up during Godzilla's initial incursion and must now redeem herself in her own eyes, as well as those of her comrades. That's a perfectly fine arc to include in the film (and one that Pacific Rim uses as well), but the script does it so perfunctorily that it lacks much impact when she succeeds.
Even the result of the film's final battle reflects that lack of follow-through: Godzilla just kind of leaves. He's not defeated, he just has another appointment, or something.