Can anyone use the JEL classification system?
I asked the AEA helpdesk. This is the answer I got:

'Any work related to economics can be classified according to the JEL
taxonomy.   The list in that sentence is not intended to be an
exhaustive enumeration of where JEL Classifications are allowed to be used.'

Therefore it's perfectly fine to use the JEL classification in a master thesis, as long as the work is related to economics.