I think you are probably
The relevance of the title is clear enough.
I'll explain the boldface words in what follows later.
I'm monumental, far from drab,
A fixture the whole world over.
Shakespeare's plays are famous worldwide.
You bet I've the gift of the gab,
Forever fashionable - moreover,
The varied globe I've travelled,
All in one remorseless city.
Ho ho. That would be the Globe Theatre.
Although the critics cavilled,
The generous still think me witty.
So they do.
My birthplace lies on a river,
Whose name makes pedants sad.
Born in Stratford-upon-Avon. I confess I haven't yet worked out what's pedant-saddening about Avon. (Perhaps the fact that "Avon" means "river" and so "River Avon" is tautologous?)
Exposure to make you shiver
And more, my rants have had.
His writings are very, very widely copied, read, and acted.
If you need a hint, in each line
Of this majestic rhyme,
Is a lustrous one of mine.
The words I've boldfaced are ones that seem to have first seen print in Shakespeare's works. That doesn't necessarily mean he actually coined them, and in several cases he surely didn't, and in some cases his meanings differ from ours. (Though in some other cases the words are Shakespearean in the sense that his are the first uses that do match our meanings.) I'm missing a couple of lines: "Grovel" has an early Shakespearean use, but someone else got there first; and I can't find anything fitting in the "gift of the gab" line. Perhaps I'm missing something.
Don't grovel - you'll know me in time.
(I don't think this adds anything much beyond rhyming with "mine" :-).)