I think it depends on the degree of overlap between the spin-offs. Dr. Who has mini-Reapers, and Torchwood has mini-Reapers that can't fly in a straight line, because it is a spin-off. I don't know much about Torchwood, but what I have seen of it, it seems to deal with a similar flavor of monsters as Dr. Who does. I don't know if the mini-Reapers appear in Torchwood or not, though.

All Stargate canons originally were assigned a single mini, the mini-Mastadge--that slimy water buffalo/camel thing from the movie. That thing might have appeared once on SG-1 and definitely never appeared in SGA. It didn't make much sense to have a single mini for three canons that didn't even have a single shared (suitable--the Goa'uld weren't) monster featured between them, and the style of monster was different in each edition.

...there is a different mini for each part of the spinoff. I guess 'focus on different characters' is the best way to put it. When I have SG-1 characters have their names misspelled in an Atlantis fic, they still get Mini-Unas, and not mini-Wraith. I don't know if there's any official ruling on how series get more than one mini type, but I think a spinoff with a different name helps. Power Rangers and Yugioh both have those, so I would assume it would be possible for them to have more than one mini-type. I dunno, that's my opinion on the issue.