Is there a difference between the durability of the chains with 1/8 and 3/32 inch? I mean, I have a single-speed bike and have chains width 1/8 inch.
The other standard I could use on my bike is 3/32 inch wide.

Is there a difference in durability?
Yes. All other things being equal a 1/8" chain will be a tiny bit sturdier than a 3/32", just by virtue of the fact that the plates are a little thicker.

I also have a single speed bike, I also choose to run a 1/8" chain.