Marking chains permanently for later identification

I would honestly keep track of them with a tag somehow off the bike vs. mark the physical chain.

However, if you really want to mark the chain, I would make a mark like a scratch on the master-link of each chain to achieve what you are trying to do. This way you know where to look to id the chain. Perhaps us a chisel tip or punch to literally make your mark.

Attach a large washer to the chain with some binding wire. Use a centre punch to mark the washer with one to several dots and keep a log of those. The washer will stay attached while you clean the chains and the marking won't be washed off by a solvent.

The only thing I can think of is to use master links from different brands (Shimano, SRAM, Wipperman, KMC …). They usually tell you not to mix master links and chains but in my experience it works great.

On older (≤8 speed) chains you might be able to put a nice, thin scratch on the face of pins with the edge of a file. On newer chains the pins don’t stick out anymore.

As Daniel suggests, additional master links or closing pins at a certain distance could be used as well.

Of course in storage you have lots of options, for example zip-ties or (electrical) cables wrapped/tied around the chain work great.

This is dependent on your choice of chain, but KMC chains are available in many colors. Wipperman chains are available in a couple of different finishes. And their master links are also available in a couple of different finishes (I've used KMC master links with Shimano chains without incident, fwiw).

Use a tungsten carbide scribe. I recommend straight lines only like ticks, crosses, and tees, not fancy letters or symbol with curves, since you may need multiple passes.