We use Mogami as an example because her final displacement was 13,600 tons when her design displacement was supposedly 9,500 tons.

How about those with green paint like the Italian and at some point the French? Is it for the same reason?

IIRC anti-fouling paint's typical reddish color at least originally stemmed from its copper content (although I'm not totally sure), but its color is hardly restricted to red (e.g. blue on USS Port Royal here).

More information on specifically the World War II Italian use of green anti-fouling paint here from shipmodels's RNfanDan.

Yes, most anti-fouling paints are usually colored by whatever main biocide is used, usually some form of copper oxide. The reason they're usually red is because underwater visibility is so poor that there's no real need to camouflage the hull (since you increase the expense by adding extra ingredients to the large amounts of paint needed). Green hull paints at the time probably included Paris green as the main biocide. Modern hull paints/treatments are mostly customer-preference for color. In recent years there's been some uproar over biocide use in hull paints (part of how most of biocide-based anti-fouling surfaces work is by leeching it slowly into the water surrounding the hull), although I favor the opinion that the minimal amounts leeched have negligible impact on waterlife, especially as most of the biocides used are already present to varying degrees in water.

Also, here is pretty much all the info you could want about IJN anti-fouling paints.