The only thing that strikes me as seriously strange here is the part about boys being allowed shoes and girls not. Perhaps the girls are doing some kind of gymnastics activities at the moment that require bare feet? Otherwise it makes no sense at all to enforce such a gender based rule.
Going barefoot for PE in general though is not all that strange unless you live in the US or Canada. In other countries including the UK and much of Europe (in other words not just undeveloped 3rd-world nations) it was common for kids to do all PE barefoot for decades. For various reasons it is now less common in Europe, but where I live, in New Zealand, it is still the norm.
In high schools in NZ kids wear uniforms that include black leather shoes. These shoes would scuff the gym floors and are considered ill-suited to running activities outdoors as well. Kids are asked to bring sneakers or go barefoot and the majority choose bare feet. Socks are not allowed because they run the risk of slipping and injuring yourself. Bare feet are condidered the optimal footwear as it allows much more traction than shoes and cost nothing. In NZ a Nike sneaker from Footlocker will start at $250 and go up.
Kids in NZ are also required to go barefoot for certain sports, such as rugby, until they get to a certain age. The reason is that being stepped on or run over in bare feet won't hurt you whereas cleated boots would.
There is no extra danger in going barefoot. Kids can kick hard rugby balls dozens of yards with bare feet. I've seen them run cross-country on uncleaned cement sidewalks as part of school PE, or playing rough games of soccer and basketball on hard tarmac, day in and day out and with not a single case of injury worse than normal cuts or bruises. Going barefoot makes the soles tougher and actually less likely to get an injury or illness.
Primary (elementary) school kids here are allowed to attend school barefoot all day anytime they want and the majority do in warmer months. They walk to and from school that way with no harm coming to them. And again, this is a developed nation, the kids all have shoes, and many even have the very expensive brand-name sneakers, but they choose to go barefoot.
Athlete's foot is caused by keeping feet in a warm moist envirnonment (like a shoe). Hookworm is only a danger if you have fresh cuts on your feet and walk in human excrement, something that may have happened in the old days of outhouses and such, but wouldn't be likely in modern America. Your hands are much more likely to bring you bacteria and viruses than your feet, but we don't wear gloves all day do we?
So, if I were you I would definitely complain about the gender based rule and the lack of option. Even in NZ we let the kids wear sneakers if they want to in all except a couple sports, the same as you'd be asked to go barefoot for Karate even in the US. It is your daughter's right to wear sneakers if she wants.
But don't be worried that this unfair rule is actually putting your child in any danger. It may be uncomfortable for her, but it won't injure her or make her sick. The taboo against bare feet in public places is a North American phenomenon. Europeans associate it with poverty and so wouldn't do it either, but only North Americans get so upset and disgusted by the idea. As pointed out earlier, all the supposed health code rules and laws against bare feet are a myth, they don't actually exist.