Oxford was a patron of a theatre troupe. It wasn't a very distinguished theatre troupe and mostly toured the provinces. However, any theatre sponsorship in the 16C helped to fund the development of the English stage at a crucial point in its history. Oxford's admirable patronage of young and interesting writers and translators, while not extensive by the standards of the day, is a greater contribution than to the arts than his poetry.
There's no evidence that he did more than lend his name to his acting troupe, however.
Shakespeare, however, was one of the first theatre workers. He acted, supported and wrote for his own company, for a changing and enduring set of players. Working for over twenty years with the Lord Chamberlain’s/King’s Men, he knew their strengths, and wrote to them.
He wrote with particular actors in mind—several times he even put their names, not the characters’, in his stage directions: “Enter Sinklo, and Humfrey, with Crosse-bowes in their hands.” When William Kemp left the Men in 1599 and was replaced by Robert Armin, Shakespeare’s vision of the Fool evolved to suit his style: not bumptious but cerebral, not Dogberry but Touchstone, Feste, Lear’s Fool.*
When he wrote Henry IV, Part 1, they must have had a Welsh boy who could sing; three plays from the 1590s** have parts playing on the comic mismatch of a pair of boys, one short and dark and the other tall and fair. And in 1611, there was even a bear on hand for The Winter’s Tale: one of a pair of polar cubs who had just drawn Prince Henry in the masque of Oberon.
*Though some scholars argue he was doubled with Cordelia, and therefore played by a boy.
**A Midsummer Night’s Dream, Much Ado About Nothing, and As You Like It.