Ordinary gear oil is designed for certain uses, chiefly heavy-duty protection for "standard" gearing, such as in a differential, and has its own series of grades which tend to differ in the amount of high-pressure wear agents etc.
Transmissions need gear protection, but many also need specific levels of friction in their fluid to work properly. Many also contain bronze synchros and so may need formulations/additives to prevent corrosion of the bronze.
Bottom line: most any gear oil will work in e.g. a rear diff, but depending on the manual transmission, some will just use standard gear oil, some will use standard motor oil (!), some need a gear oil designed for manual transmissions. Unless you've researched it pretty carefully, I wouldn't change from what the manufacturer put in. The types of problems you run into tend not to be immediate, but rather needing new synchros in 15k miles or things like that that aren't cheap.