I was surprised to see that you don't get any certificate, he told me it's all online.
Well, yes and no. Yes in that it is online for the doctors to see. But no, in that the pilot is issued a document which has the medical printed on the inside for the first issue, after which a stamp is placed for subsequent medicals.
Now I need the medical license number they call it (TCCA pilot file 5802 number) to complete my application but I don't have any.
The 5802 number is simply your license number (well, actually the numbers after 5802 is your license number); however, because everyone involved at Transport uses this number for various things, it ends up having multiple names like "medical number," "license number, "file number," "5802 number," etc.
It is a six-digit number that will follow 5802 (e.g., 5802-123456) and that will be printed on all official documents you receive from Transport, normally after your name and address and either in the top left or top right of the first page.
If you can't find the number on any documents you've been sent
after completion of your medical, you can always call the local Transport office that you're using to convert the license, and they can let you know what the number is, assuming the paperwork from your medical was submitted and processed already.
In your case, the license number will be generated as a result of your medical, so if it has only been a day or two since you saw the doctor, it may still be making its way through the system.
Remember too that it is still close enough to Christmas and New Years that many government employees are only just starting to come back from their holidays. Indeed, the 18th of January is typically taken as being the first day of the year where Transport is back to full strength, so if you did your medical quite recently, the paperwork might still be sitting on someone's desk(top).