I can confirm this too, but it seems like the ability to take other clothes depends on whether you are the host or a client (non-dedicated server).

The host is able to take other clothes, but clients are not.
It seems like this is the case for the cargo items, like shirts, carrier rigs, tactical vests, backpacks and so on. Helmets are not affected by this issue since they are no cargo items.

What we did:
We created an example mission with an ammo crate which holds some clothing items. The host can drop his default cargo items and can take new cargo items from that ammo crate. Clients can drop theirs, but can't new ones from the crate.

I'll try to upload a very small example MPMission called "clothingTest.Stratis.rar" here with said ammo crate.

Can confirm this. Would love to see this get fixed soon for a "prison escape" mission I'm working on. :)