Both my fiance and I use custom PNGs for skins in multiplayer that we downloaded from minecraftskins:

Me: https://www.minecraftskins.com/skin/13617664/sora/
Her: https://www.minecraftskins.com/skin/13766199/yoshi/ (likely)

If I join the server and we are a large distance from each other, the game shows the correct skin for us both. However, if we move into the distance where our characters are visible to each other, the skin swaps to what the other sees. Once the skin is swapped, we can no longer see our skins in inventory or in third person, but we can see our skins in the model when we hit escape.

There is an obvious bug with skin loading that you've recently added in either 1.14.0 or 1.14.1, as we have only recently noticed this issue.

Note that if I use one of the new skin models that I have to build, and she continues to use her custom skin, it does not swap textures.

We have tried deleting all other profile characters as well, to no avail.

Ignore the resource pack issues with some of the screencaps, we disabled all resource packs and the problem existed with or without them.