Will #20 mono and #10 mono have the same stretch or will one have less, assuming it's the same brand.

Thinner line will have more stretch all else being equal. 

What do you plan on using #20 mono for?

I ask because I use #20 big game on my swimbait combo for throwing baits up to 3oz.

I throw heavy swimbaits on 25lb big game. Not much stretch

Thinner mono stretches more

Copolymer line has less stretch. I'm happy I switched.

I'd Go with Excalibur silver thread copolymer line.

I think 10 and 20 lb lines of the same mono will stretch the same before breaking.  But the 20 lb mono will take twice as much force to begin stretching.

By "thinner" I believe they mean smaller diameter or less pound test.

Every plastic stretches and metal for that mater, it's called yield strength, point where enough force is applied to yield or stretch the material. What anglers sometimes call stretch with nylon monofilament is the force where the line yields and starts to elongate. If you use about 1/3rd the line pound test as the force required to elongate nylon line you should be safe. 20 lb mono nylon line yield around 7 lbs of force, 10 lb nylon mono line about 3 1/2 lbs of force. Fluorocarbon elongation strength or stretch is about the same as nylon mono. Braid is much stronger and yields about 3/4 it's pound test, it does stretch before it breaks.

Tom

PS, by setting your reel drag at 1/3 rd the line strength you avoid line stretch!