This is a fairly easy question. My character lost most of her possessions and supplies, including her packed clothes. Assuming they're of a similar height and build, would there be any issues with her wearing clothes borrowed from a male friend?
Depends on the type of clothing and the character's build. If she has a slim, "boyish" figure I don't see a problem with her wearing men's clothes. If she's curvier and the clothes are fitted, they'd probably be baggy in some parts and too tight in others. And depending on chest size, she might miss having a bra.
If it's modern or semi-modern, you might give her a moment of fumbling with a buttoned shirt or jacket, since the buttons are on the opposite side of men's to women's clothing. I personally know this because Mrs. A and I have, by accident, matching-pattern flannel shirts. Yeah, it felt tight, but I was still half-asleep and could not for the life of me figure out what was wrong with my fingers that morning...
I don't see any reason why not. I've had female friends and roommates borrow my clothes plenty of times. Mostly sweatshirt and stuff, but my old roommate and I used to share clothes all the time. If they're intimate the amount of sharing goes way up, my wife even likes to wear my boxers sometimes. What time period / culture are you in? She certainly couldn't do it in Saudi Arabia, but in western cultures it's common for women to wear men's clothes. Assuming the mans shoulders aren't too wide or her breasts aren't too big.
Shirts can be easily swapped. You might run into problems when it comes to pants, but say they're both pretty skinny and her hips or butt aren't too wide, and neither of them are into skinny jeans (unless very stretchy), they could wear each others' clothes. Sometimes girls' shirts and hoodies have too short sleeves for guys, though. And sometimes men have broader backs, but it tends to depend on whether they work out or not, while women's shirts are often pretty narrow around the shoulders. I sometimes have to buy S instead of XS because of this, otherwise I can't move my shoulders or arms freely. I'm not sure if this is a problem in the M-XL range. But since your girls is borrowing clothes from him -- no problem.
Since you asked about issues, the above explanations have you covered regarding fit. The other thing I thought of is scent. Is the guy heavy on fragrance or deodorants? Because there's a small risk, depending on how thoroughly they're laundered (assuming she's taking clean clothes here), that she might have an allergic reaction to any chemicals in his clothes. Usually, this will be a light rash and/or skin peeling or burning. The same for him if he gets his clothes back from her. But that depends largely on how exposed they are to those chemicals. Women wear their boyfriends' and husbands' clothes a lot. I don't think that's normally a problem, but again, it depends on those above mentioned circumstances.
Men's pants don't account for butts & hips. As a confirmed tomboy in my younger days I experimented with wearing men's pants and found that while the increased number of pockets is good, they are either too tight on the hips or too baggy on the butt to be truly flattering to a woman's figure. And I say this as a slender woman with narrow hips and a small butt - the pants fit in a technical sense and that's about all. It would be much worse on a busty/curvy woman. Shirts are less of a problem. Again too big most of the time, looking more like an oversized nightshirt than anything. A busty woman could pull it off without looking like a dude, but if your character is less endowed in the breast department she'd look more tomboyish. Depending on her personality she may not be too happy with that, even if it's better than nothing, whereas a woman who values practicality over looks might be perfectly okay with it.
There are other subtle differences other than size and shape. Male & female shirts and jackets usually have the buttons / fastenings swapped left for right. Male shirts, especially workshirts, tend to have more cuff buttons to make it easier to roll the sleeves up. Womens' trousers often have shorter flies (can't think why ), or a zip at the side. You say 'assuming they're of similar height and build', so there shouldn't be any problem other than she might be a bit fumble fingered getting dressed and undressed. Your female MC might even find herself becoming increasingly attracted to her male companion as his pheromones, absorbed by his clothes when he wore them, work on her subconscious libido during her most fertile period.
