Allowed by the 2012 rule change. Nothing I've read on the subject indicates that the Egyptian team received any special disposition for their uniforms other than being allowed under the new rules. I'd be interested in seeing a citation if that is not the case.
It's funny, The same year (2016) that Egypt showed up in their uniforms, April Ross and Kerri Walsh Jennings had their sponsors design a unique uniform for the same Olympic Games. Four years after the rule change, not only was the bikini still their choice, but they wanted a better version.
Their choice.
It's not a great source but:
https://www.stuff.co.nz/sport/olympics/82926043/evolution-of-womens-beach-volleyball-uniforms
but the key point here is this:
The FIVB said the change was made in order to "respect the custom and/or religious beliefs" of players.
which is choice for all the wrong reasons. Players can now chose the original format (which was very specific and explicit about how much flesh can be covered up (not much), an intermediate option of t shirt and shorts or a full body covering, but the reason driving the change wasn't feminism any more that the reason for the skimpy outfits on Trek was either.
Roddenberry (a man) wanted women in skimpy clothes in 1966 whilst Islam (a patriarchal religion) wanted women covered up in 2012. I get that the rules allow for choice but the idea that said choice is the bedrock of feminism starts to look a bit suspect if said "choice" is there in order to accommodate another form of repression,
Kerry Walsh may well feel the way she does, but she doesn't speak for the sport. Someone called Anna Scarlett apparently gave a very different opinion, but the feelings of players such as herself were not, it seems, the driving force behind the change. Instead it was respecting religious sensibilities.
Good move but for all the wrong reasons, just like Trek.