If someone proposes approval voting be used for a many-winner election, I'd assume they mean "leave the single-winner district system unchanged and just delete the rule from the ballot that says 'choose only one candidate'."
But any system with single-winner districts will be nonproportional and will tend to deliver more power to parties that are one of the two largest (more precisely, the largest in each region, sometimes the whole country). As with all district-based systems, everyone who doesn't vote for the local winner is basically ignored, and this can be more than half of all votes.
I think this is a good explanation for why, if a major party proposes electoral reform, they tend to want Instant Runoff Voting (a.k.a. Alternative Vote) instead of any proportional system. IRV and approval voting (in single-winner districts) both preserve the big party's tendency to win more seats/power than their level of approval relative to other parties would justify. (Another possible explanation for this is politicians having a severe ignorance of the space of possible voting systems, plus the fact that IRV has been, for whatever reason, a popular proposal in some circles, e.g. Ralph Nader harped on it. But in Canada, when the politicians on the ERRE eventually chose proportional representation, the prime minister made up an excuse to ignore its findings and break his promise that "We are committed to ensuring that the 2015 election will be the last federal election using first-past-the-post.")
As Marcus mentioned, there are versions of Approval that are ostensibly proportional, but I don't understand what "proportional" means in this context. If I vote for 5 of the 10 candidates and you vote for 1 of the 10, I don't know what the "right" way is to weight my ballot relative to yours, except for an intuition that my ballot shouldn't be more or less "powerful" than yours. Marcus mentioned SPAV, which strikes me prima facie as not proportional in any obvious sense (and the CES's description of it does nothing to reduce my concern.) PAV sounds vaguely better, but I don't know why one would choose it over another multi-winner non-party-based system like Single Transferable Vote or Simple Direct Representation. (edit: Wikipedia says PAV 'satisfies a strong fairness property called extended justified representation'. The STV page doesn't list any formal properties that STV satisfies, but Wikipedia does offer a table of properties of multi-winner systems.)
Thanks. These algorithms seem like they would be better for passing the independence of clone alternatives criterion.