“Did” is the past subjunctive, used to talk about an unreal present. “It’s” means “it is”, which is present indicative, used to talk about a real present. The grammar is the same as in Portuguese: Já é hora que a gente eliminasse a monarquia. This unreal present is related to the “second conditional”: It would be great if we did away with the monarchy. Seria legal si a gente eliminasse a monarquía. I wrote a surprisingly popular mini-article about conditionals in English: Grammar boost: Conditionals cheat sheet PART 1 https://www.italki.com/discussions/1356999

This is a subjunctive sentence. The word that is implied but not actually written. This is why did is used. You could use do, though it would be rather unnatural. 'It is time to do away with the monarchy' would be better.

I really like this question because it forces me to think. But I would say there really is no difference. I mean, you could say that "it's time we DID away..." is a little more intense. But I would say it makes little to no difference. "It's" is always "It is" or "it has" in sentences like "It's (it has) been a great day," "It's (it has) worked out alright". "It's" will never be "it was, " at least in American English.