Learn English – Word order: “dear the lord” in Shakespeare

I think there is a clue to this just a little farther in the passage itself, where the speaker goes on to mention his "state"... and I think it is in the notion of "statehood" that these expressions (and perhaps their understanding) are to be grounded.

That is, the speaker is speaking from the state of not just being a man, but man, as in ALL men, and further, ALL (of) "man", that is, a state of mind not just identifying with all the members of that given set, but to represent them singularly, collectively, and superlatively (cf. expressions like "I am woman, hear me roar!").

Additionally, there is an allusion to "man" being used here as an abstraction, not just a concrete representative of same, or all of them, but the whole idea of being a man (or woman, later), as in expressions like "I am VENGEANCE," or similar identifications with abstractions of concrete things, or abstract concepts themselves. This identification-with-the-concept is what gives the speaker the presumed mandate to speak on that entire concept's behalf, as he indeed goes on to do throughout the quoted passage.

However, as the somewhat more modern examples above suggest, I am not sure this is so much a feature of Shakespearean English, but rather, that Shakespeare's prose simply uses more of this forceful stuff. Indeed, I would bet you a shirt frill that if H.G. Wells dumped you in ye local publick house in Merry Ole Stratford-Upon-Avon, you'd be hard pressed to hear any of the locals express such prose while quaffing their brews.

But that, of course, is pure speculation on my part. :o)

No, Shakespeare's double negative was not grammatically wrong. According to David & Ben Crystal, the rule that two negatives make a positive was not applied to most uses of language:

[T]he strict mathematical logic was used only in a few formal styles of expression.

A double negative was just a way of intensifying a negative. (One could say that this usage closer to the mathematical formula (-x) + (-x) = -2x.)

The idea that using a double negative always results in a positive became dominant through the works of prescriptive grammarians in the 18th century, for example Robert Lowth's book A Short Introduction to English Grammar (1762). (At least, this claim is made on the page Early Modern English (c. 1500 - c. 1800) on the website The History of English, which also mentions a few similar works.)

David & Ben Crystal also point out that Shakespeare even used triple negatives.

(See also my website for more examples of double negatives in Shakespeare's work.)

Update: curiousdannii pointed out in a comment that this is also known as negative concord. (The term is also mentioned in the Wikipedia article double negative.)