There are many ways that the registrations could have been found and here are most of them.
On the original full resolution file, the registration of the higher of the two planes (B777-200) is likely to have been easily readable on the actual shot!
With ACARS it is relatively easy to find the registrations of flights.
It is possible to get lists of movements with registrations for many EK
As already stated, the photographer was on board an Emirates flight. He could have noted the registrations of those aircraft on the ground as they may all have departed Dubai within quick succession, and if they were all routing towards Dubai, then all three would have landed within reasonably quick succession.
Indeed the crew may have been in communication over a company frequency and noted the registrations there and then.
If the photographer was a member of staff of Emirates, again it would have been easy to look up Emirates or Dubai Airport internal documents or intranet.
Registrations can be ascertained easily from Selcal codes used on HF