On the AS400 where my iWay server is installed, for each table there are several masterfiles names as follows:

EDAMAS.table
EDAMAS_53.table
EDAMAS7.table
MASTER.table
MASTER_53.table

I'm guessing the _53 ones run first, and EDAMAS and MASTER last, but what is EDAMAS7 for? And why is there no corresponding MASTER7?
Why so many?

You can have as many master files for a table as you want, why you have so many is a question for whoever creates them on your system.

Are they actually master files i.e. their extension is .mas?

I would suggest perhaps from the names that they refer to WebFOCUS versions. Did you have version 5.3 and then upgrade to version 7?