Do app icons change when Android does a system update?

not necessarily. it will depend on the app and its developers.

 

Sometimes change, sometimes stay the same, depending on what the manufacturer of your particular phone has pushed out, and how they might have customised their Android 10 update, specifically the icon set and launcher it uses.

 

Depends on the app.

 

System app icons may or may not change. Most likely is that only some of them change anyway.

Third party app icons do not change, since the update doesn't change those apps at all. Occasionally a system update may impose a style, e.g. masking all icons so that they have rounded corners, but that's unusual.