Until Disco S3, “Living witness” was the furthest future we had seen in Star Trek. But Academy takes place after Discovery. At the end of “Living Witness” the Doctor is described as heading towards Earth.
Assuming the burn didn’t get them, it’s entirely possible two EMHs are surviving in Discovery’s time. I think it would be a fun twist if it wasn’t the EMH we expect!
After a thousand yeas of vastly different experiences I would be surprised if they were very similar.