Mower oils are better in your mower than car oils.
However mower branded oils are no better than plain old mower oil
So generic mower oil will be fine.
One day mower companies will wake up and sell their oil at a competative price.
I use oil from my mower parts wholesalers in all the mowers I service.
You can use any oil that meets the SAE specifications for your engine. Air cooled engine oil has an additive package that is designed to operate at a higher temperature, than water cooled engine oils like what you put in your car. Just like you can use regular oil in your ATV but the friction modifiers in auto oil can cause the clutches to slip in wet clutch applications.