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I can't see any advantage in having the same chain on all your bikes.
A standard roller (bushed) chain would be preferable to a bushingless chain though.
However there aren't many bushed chains made nowadays. There's the Connex 7R8 (3/32", 1R8 is 1/8") and KMC B1 (1/8"). I'm sure if you roused Tstr from hibernation he could tell you others.
Having said that, apparently you're still living so I suppose it's fair to say that while one is technically better than the other, it's not critical. Personally, I'd choose a bushed chain, over matching the width to the sprockets (the 7R8 is "nice but pricey", the B1 is very reasonable).
TLDR: whatevs.
Time to change the chain on my Day One.
Currently fitted with a kmc (I think) 8 speed chain but every other bike I have runs a kmc 510 (1/8" singlespeed chain).
Do I switch to a 510 on the Day One so it's the same as all other bikes or stick to an 8 speed chain.
Chainring and sprocket on the Day One are obv 3/32" so I know some of you will say a 3/32" chain will run better but most of my other bikes have a 3/32" cog at at least one end and I don't have any complaints of how they run.
TLDR: 1/8" or 3/32" chain for singlespeed?