Motor oil requirements on exhaust system

I have been working with 2-stroke engines for quite a while with a motor oil that has doing a great job. It was kinda simple since there was literally only 2 different oils at the gas station for 2-stroke. Recently I’ve gotten a Honda GX-35 engine which is 4-stroke. I have been going to my local gas station as well and noticed that there is a plethora of different oils (same brand, different types). While it was quite obvious which one is the right for my small GX-35 engine (described as motor oil for small engines like generators, light diesel or gasoline engines) I was wondering about some of the descriptions of the other oils.

The point that struck me most interesting is the fact that the different oils had different statements about the exhaust system you should be using for that oil. For example one oil mentioned, that it is very light for high performance but requires a very modern exhaust cleansing system and is not compatible with diesel particle filters. So this made me think: What causes this different exhaust requirements? Do some oils not fully burn out or burn to particles that could be toxic (thus modern exhaust) or not burn up in the particle filter?