the independence of clones criterion measures an election method’s robustness to strategic nomination. Nicolaus Tideman was the first to formulate this criterion, which states that the winner must not change due to the addition of a non-winning candidate who is similar to a candidate already present.
That doesn’t seem to be what the OP is concerned with at all, nor does it appear that Approval would violate this criterion.
The problem you’re pointing to is called Cloning. The Electowiki article on approval voting has a short section about multiple winners. What you’re looking for is a Multi-member system that passes the Independence of clone alternatives criterion.
Weirdly, because I think you’re right, non of these pages mentions the clone problem with approval voting.
From the link:
That doesn’t seem to be what the OP is concerned with at all, nor does it appear that Approval would violate this criterion.
Reading the Independence of clone alternatives page and the Strategic nomination page the principle seems to be more general than the clone page (from which the quote comes) makes it seem..
Thanks; this language and these links are very useful.