Say it is 1590 or 1600 (or whatever), and I am talking to someone who is overly flirty, or taking excessive care with their word choice or rhythm, would it make sense for me, in that moment, to say something like "Okay, cool it Shakespeare."?
I guess this also begs the question on whether people predominantly spoke like Shakespeare at that time. I vaguely recall learning that his writing style was more or less the vernacular of the time. With that knowledge, if someone was holding a regular conversation and sounded very similar to Shakespeare, would it be out of the ordinary or tease-worthy?
Shakespeare was not among the more mannered or rhetorically excessive writers or playwrights of his day. John Lyly was the iconic composer of pretentious and mannered prose, in the Elizabethan period (i.e., from the point of view of the period itself). His style of prose was extremely popular (and hence well-known), for a time, but ultimately discarded as a self-indulgent courtly absurdity, which receives very little attention today. And this style was parodied by Shakespeare himself. This characterisation does not extend to Lyly's drama, incidentally (which is still appreciated in some measure). But it was his prose which made an impact.
However, if an individual were to identify a name with this style of rhetorical excess, it would likely not be Lyly himself, but Lyly's protagonist, Euphues, out of whose mouth comes all this nonsense. Euphuism had a sufficient cultural and artistic impact that "that's enough, Euphues" would be a sufficiently comprehensible jibe, anywhere among the literate class. By comparison, it's hard to see what "that's enough Shakespeare" could possibly be understood to mean.