The two main proposals are sequential proportional approval voting (SPAV) and proportional approval voting (PAV).
SPAV proceeds in rounds. In the first round, the candidate with the most votes wins. In the second round, the ballots are reweighted such that those which have the first winner selected have 1/2 weight and all others retain full weight. This is repeated until each seat is filled, and, in each round, a ballot that has voted for n candidates who have already been elected is weighted at 1/(n + 1).
For an election to fill N seats, PAV looks at each possible set of N candidates and elects the set which maximizes the utility function given by 1*(# of ballots with at least one of the candidates selected) + 1/2*(# of ballots with at least two of the candidates selected) ... + 1/n*(# of ballots with at least n of the candidates selected).
Both PAV and SPAV yield proportional representation while using approval-style ballots. PAV yields better results, but SPAV is easier to explain the outcomes of. Another option is to use the allocated score algorithm on approval ballots.
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:
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.
There is quite a lot of active research in computer science about this topic. Here is a recent survey: https://arxiv.org/abs/2007.01795
PAV has already been mentioned, and is a very good rule if you want to ensure proportional representation. Its sequential version is not known to guarantee any representation properties, though in practice it might do well. There are some other good rules around, including Phragmén's rule, which unlike SeqPAV provably satisfies representation guarantees. Like SeqPAV, it produces outcomes that can easily be explained.
Regarding your question (2), the proportional rules I mentioned may in the worst case return a committee whose utility (i.e. utilitarian social welfare) is a 1/sqrt(k) fraction of the highest possible utility, where k is the size of the committee. See Section 5.5.2 of the survey I linked.
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.)
Per Wikipedia, the answer to your first question is "no, that isn't how approval voting works if used for multi-party elections."
My personal solution to the voting system dillema is candidate directed instant runoff:
This is just a variation of asset voting. I like it too. I could see an argument that you should start by redistributing the votes from the people who are guaranteed enough votes for a seat, because that could change elimination order. There are a bunch of different heuristics you could use.
A lot of voting problems are actually group-preference ambiguity. For a given population with static preferences and independent positions (that is, the selected body is just a list of the top N candidates, not a slate where less-preferred individuals become preferred as a group), I'd expect that cloning is ideal.
The populace would actually prefer N copies of the best candidate, rather than the N-1 not-as-good people.
In the case where the interaction of selected options matters, and for instance, all cooperative second-best party is preferred over a mixed-party of the best people, then this kind of vote fails to serve regardless.
It's not at all clear this is a problem. If all the winners are the closest to the centroid, then you will have statistically about the same overall ideological center within the group regardless of whether a proportionality is used. You might expect the lack of different perspectives to cause a problem, but a bunch of centrists can solicit expertise from multiple perspectives. Which makes sense since they are vying for every vote. Not to mention that a body of centrists will tend to get along a lot better than a bunch of quarreling extremists like American leftists and Trumpists.
But as was noted, there are things like sequential proportional approval voting if you really want PR.
Suppose a nation is electing k individuals, to populate a senate or something. Now approval voting has some good properties, so suppose the protocol is to let each of the m citizens vote for as many candidates as they want. We then award a seat to the candidate with the most votes, and deduct m/k from their vote tally, or thereabouts. And then we repeat the process using the new tallies, until we have k elected candidates. It seems like a problem with this approach is that if the most popular candidate could copy themselves k-1 times, then their copies would all have the same score as themselves, and so they would completely monopolise the senate. In a very real, practical sense, this protocol would mean that one political party could dominate the senate. This would undercut the ability of the senate to combine disparate views, and to represent the diversity of opinions of the population, as well as reducing its robustness to takeover by one group.
My questions are: 1) is this how approval voting is proposed to be used for multi-individual body, and 2) is there any suitable proposal that would retain desirable properties of approval voting (simple interface, and attaining high utility) while avoiding this problem?