Do all 10-speed chains fit 3x10? I'm wondering if all 10-speed chains fit to 3x10. A lot of webshops say that the chain is "for road bikes 2x10" or something like that. Or nothing at all.
The chain I have right now (Shinano CN HG-54) is "ultra narrow", but I'm not sure if that's because it's a 10-speed, or if there are sizing issues between 1x10, 2x10, and 3x10.

I appreciate any input!
10 speed works with 10 speed. You will be fine with the chain you have. The major difference being that the one that is made for 3x10 will be a couple links longer generally. Usually this is designated by a number such as 116L where the L stands for links.

Width wise you should be fine with any 10 speed chain. 10sp chains are slightly more narrow than a 9sp so that they could pack that extra cog on the cassette. Although when you look at the difference it is minuscule.