As has been said, the EVSE has its own GFCI functionality built into it, so does not require a GFCI receptacle to work properly and safely.
Older receptacles often have lost all of their spring tension on the socket. That's a good reason to replace them, but a loose connection could be giving the EVSE issues.
In the scheme of things, many things don't need the safety ground provided on the third pin of a US style plug. That pin should never have current on it except in a fault, and is therefore, essentially unused. From an ultimate safety viewpoint, a basic 2-prong device is relatively safe. Adding a ground pin, MIGHT make it safer, depending on how it is built. Plugging in to a GFCI protected circuit is safer, and it does not require a ground to function - it's looking at the power - everything that goes out one lead MUST come back through the other, otherwise, there's a ground(ing) fault...i.e., some power is being drained off to ground, maybe through you...the device doesn't need a ground to detect this at all as it is exclusively looking at power/neutral. That protects you, an arc-fault device protects the wiring and ultimately, your property that you might be in at the time, from a spark induced fire.
2011 535i x-drive GT, 2014 i3 BEV