I got some finding on it. (@V1.14.30)
My son, my wife and I are playing Win10 version, we face the problem of characters switch during local LAN multiplayer, you are mine, I am yours, it happened over two months. All three of us using custom skins.
But after troubleshoot, we found that all our custom skins are using the thin body. Today, I tried to change my son custom skin to thick one, keep my wife with thin, and I purchased the new skin. Then the problem solved, all we can use our own skin and when we look at each other we can get the right characters. The preview when press E also correct.
After that I changed back my skin the custom with thick body, we still can look other correct, but when my son (thick body) and I (think body) press E to check, our characters exchange. But my wife does not affect.
The mash up of characters look like due to the body type, if you have only two in the local LAN, I recommend you use difference body type to custom your skin, more then two, sorry, please purchase the skin pack or wait for new buy fix.
Besides, create your own skin pack also is a good solution for us as mention by other player. I tried today and not difficult.
https://www.youtube.com/watch?v=rrsw9GyP4Ro
We may follow the above, make your own skin pack, everyone select their owner skin pack when play in multiplayer mode. The skin mash up of characters gone.
Mine are player WIN10 1.14.30