Standard Temperature and altitude

when a manufacturer says the engine produces a maximum of 300bhp, are there any SAE standards that say it was measured at certain temperature and altitudes?

It depends who is advertising it and what country you are in. In the US there is no regulation on advertised horsepower. Manufacturers don't have a regulatory organization overseeing claimed power like they do with fuel economy. Other countries like China have specific rules and more regulation on power claims. The big 3 automakers in the US (and their associated brands) voluntarily certify horsepower according to SAE witness testing. There are definitely some loopholes and wiggle room the manufacturers have to get the power numbers they want, especially with backpressure and intercooler efficiency.

To directly answer your question, from SAE J1349 the standard SAE temperature is 25C at the airbox with 99 kPa supply air pressure. This is basically the temperature and pressure on a June day in Detroit. The correction factor are supposed to be within 3% to be considered, but this is frequently ignored in all but official certification tests. Dyno shops use the outdated/no longer valid "STD" correction factor, which is for small engines and comes from SAE j607. Please see attached documents.