I've never encountered an Island like that, which are the ones you are referring to?
Cause I've gotten multiple default islands, sister fruit, hardwood, money rock islands, bamboo islands, scorpion and gold rock islands on the same days before.
Only one I haven't been lucky to find is Shark-Fin Island.
The islands all have a percentage of encountering them, nothing is locked to one visit per day. It's just odds.
The only restricted islands I can think of are the time sensitive ones.