Making a guess from the info that I can remember about the data retrieved from a BMW remote key...the answer is probably no.
The data retrieved from a BMW‘s remote key (from my memory) are things like:
Vehicle Identification Number
mileage
fuel level
last time driven
CBS (condition based service) info like when the next service is due or past due etc. (this is the info that can be found in iDrive under Vehicle Service etc)
(newer keys may also have the capacity to store GPS data)
If this message is stored...it is probably stored in the ECM (electronic control module) related to check control system. If you have software that can retrieve data from the car’s ECMs...you may be able see that information if you connect to the OBD connector with appropriate hardware.
Since you asked the question...it would be interesting or revealing to know WHY you asked?
It doesn’t appear to be anything worse than getting a check control message that your washer fluid is low, or that your trunk is open etc. If you add the oil, washer fluid, or close the trunk...all is well.
EDIT:
You can see in the video (at the beginning and at approx 42 seconds into the video) the info that the key reader gets from the key. And again, the CBS info is more about status of the various systems...instead of listing when you were low on oil at a specific mileage or date. Specific info like that is probably stored in one of the ECM (electronic control modules) that records and issues the check control status.