The main reason is that having the tags on the page for the game makes tagging much easier. They should also be sorted by some logical order rather than alphabetically. Take post #5021088 and post #5107193 for example. I could tag fire emblem: three houses, use "related tags", and click on the characters that are in there. Easy as that. Can't do that when they're instead on the wiki page for something that isn't even tagged.
There's also just no real reason for the characters to not be on the game wiki pages. Yeah, it makes the wiki pages long, but their whole point is to provide information. Without them, most of those pages are barebone and nearly pointless.