Technically it is 90 days (not 3 months) that you're allowed to stay while waiting for the process to be done and your visa is processed - this is technically the Schengen short-term stay visa.
It makes no sense for the 90 days to exit the country count after the visa is issued, since you'll have a long-term stay visa and no need to exit the country.
Note that you may be ellegible for "switching" your standard Schengen short-term visa for another type of visa (etudiants, vacances travail...) depending on your circumstances - more information here
After clarification of which country OP needs to exit: if your visa requires a switch to a residence permit, 3 months is the time you have to report yourself to the prefecture and obtain your residence permit. If you only require validation, that's the time you have to validate your VTS-LS. You're going to see that your long stay visa has an expiration date, it counts after it is issued.