Among other things, I'm also a landlord, in a city turning a sharp left turn in 2020.
Making electrical changes like the one you are talking about is not allowed, except if your landlord approves it in writing and the work is done by a licensed pro.
You don't know how packed the main panel is or if there's any space for added amperage.
It should be mentioned in your RENTAL AGREEMENT. Read it if you still have it.
My policy: I would go along with a trouble free, loyal and respectful tenant's request and would allow an extension cord if the shed is less than 20 ft away and put something in writing.
In regard to the ownership of the shed, I would let the tenant take it if it's portable, when he moves out. After all, I can build my own shed if I so desire.