This isn't about own vs rent, it's about house vs apartment.

Open flames are dangerous and smoke is annoying to neighbors. Condos and coops typically won't let you grill. Some of them have designated grilling areas and those often have restrictions on how you can use them. Even many apartments with fireproof balconies won't allow them because not all the neighbors want a balcony full of smoke.

Every house I've ever rented, allowed grilling. Even the cheapest one, a row-house in Baltimore, let you grill in the back "yard".