What to do if I don't have dental implant records and tooth is loose?

The dentist is right to want to know the manufacturer.  Sometimes we can ID via xray, but unless you KNOW the company, you may need to pay a fee for the dentist to open the crown/implant and see if their parts (screw drivers, etc) fit.  THEN new parts can be ordered.Knowing the company in advance can save 1-2 visits (thus, money).There IS a website that helps narrow the guessing.  Google search "What implant is this"

Dont worry, xray will help in identifying the type of implant placed. There are some paid simulation apps available with which you r implantologist can identify the type of implant placed. 
1. Best option will be to try to communicate with your old dentist who has done the implant or who has restored your crown. Usually the doctor have a record.
2. If the crown is a cement retained crown, the implantologist will have to drill a hole in the crown to reach the screw blindly by correlating with the X-ray , then multiple types of hex driver can be tried to open the screw which connects the crown to the abutment.
3. If the crown is a screw retained one - (with a hole in the centre visible on the top of crown for back teeth, or inner aspect for the front teeth)  , the resin will have to be removed and by looking at the xrays design of implant can be expected and multiple types of drivers can be tried to open and then change or tighten the loose screw. 
All the best

This is always challenging as there are so many manufacturers of implants these days.  You may want to consider seeing someone in your area who places a lot of implants and who is familiar.  It will still be difficult to identify.