I've never had a problem with starting fluid damaging an engine when I used it when needed.
However you should not use it to try to keep an engine running because that could cause a fire and could damage pistons etc.
But just spraying some starting fluid into the engine to get it started when it won't start normally is okay and will do no damage to the engine.
The starting fluid is meant to provide a boost to get the engine started and when used properly should do no harm to any engine.
It's not recommended to use the starting fluid in diesel engines though because the starting fluid can do more damage to diesel engines.
But still people use the starting fluid in both diesel engines and gasoline engines without any problems.