I am, of course, quite conversant with the expressions "the likes of him," "the likes of them," or "the likes of me."
"To my like" sounds just... off. I genuinely thought it might be a typo for "ilk."
But he's the world-famous playwright, so who am I to argue, right?
Regards,
It certainly isn’t a common usage, but neither was “bigly”. Had never heard that one before Pres. Trump used it.