The bagger is not necessary on your Murray.Murray is no longer in business,but you can still buy a side chute that will allow the grass to safely discharge.
I would imagine that no other bag would fit on a murray. Every brand makes attachments that will only fit their brand and no other so unless there is a generic knock off of the brand you probably have to use a murray.
There are many aftermarket baggers being made for different mowers so the answer to your question is no. Modifying a bagger to fit is fairly difficult though.