FWIW, I doubt that Dr Johnson is having a go at Shakespeare. From what I have read, Johnson was quite transfixed by the bard, ever since he first read "Hamlet" at the age of nine. He studied the plays in enormous detail, using quotations from them to illustrate about one third of his dictionary definitions. His objectivity is obvious - he was quoted in 1769 as saying,"Shakespeare never has six lines together without a fault" - but the sentiments expressed in the "poet-ape" piece do not correlate with the overall respect he seemed to have for WS.
He was particularly taken with "Macbeth" and reputedly quoted lines from the Scottish Play on his deathbed.