Why the constant use of, e.g.? As part of your repatriation efforts you should learn how to use English properly.

So Shakespeare mentioned the recorder once or twice, certainly not as often as he mentioned other instruments (no e.g.'s, sorry). No reason at all to suppose that he did. If Shakespear played anything, I suspect it would have been some string instrument so he could serenade girls. Bill was far too busy to take up music seriously, running a show and writing takes up all your time!!
As for little organ, it is not certain that he meant the recorder, merely some small instrument (organ meaning instrument, not the pipe-monstrosity in cathedrals).