Well in Phantasy Star things went as a series until Phantasy Star 4, and that is what makes PS one of my favourite series (and PS4 tied with Chrono Trigger AND FF6 as favourite RPGs).
I don?t considere FF, Mana, Tales, BoF or Wild Arms (even though this one says it is always in the same world, no concrete links are presented, at least between the ones I played). And I still find it lame regardless of the success they make. Call me whatever you want, but I perfered times when companies where smaller and while being on business, you could see that the team was more involved in the projects, including marketing than the cold way things are now (though it goes beyond game market).
Ans yes, I may not like their marketing politics, but still like some individual tittles like FF6, Secret of Mana, Tales of Destiny, Tales of Phantasia, Wild Arms 1 and 2... just don?t like the series they belong to, or rather, won?t considere a simple trademark as series. It would be like naming Jurassic Park as Star Wars Episode 7 and considere both a series.
And yes, prequels sequels (and not forget side stories) can actually be really good, and Chrono Crisis IS a lovely example, in fact I find it more worthy of the Chrono tittle than Chrono Cross (which had ties with CT, but still felt like something set in a completely different universe).
As for tribute games, Marcus said everything and I agree to that. I still think an FF tribute game can be good (FF Endless Nova brought me back to rm) but still feels like the creator is under the shadow of their loved game rather than making something to pay homage to it.