In TNG "chain of command" part 1, when Picard transfers command of the Enterprise to Jellico, he reads the order and states that he is requested and required to give up command. Why the "requested" part? If he's required to do it that seems like it's unnecessary. Is it possible to be required but not requested?
This phrasing is straight from The UK's Admiralty orders. So yes, it is an anachronism, but it is also a very functional and precise wording. This is an homage the TNG pays to the wet navy times, which would imply that StarFleet upholds the best traditions of the past. Of course, really it all depends on the general naval knowledge levels of the writers, who are, in all of Star Trek, sometimes so egregiously off that it's cringeworthy.
I think Stewart might actually be the one to suggest it - it would be a sort of "Shakespearean moment", and very well placed...
As for the wording itself, it is an old Royal Navy custom. It is, on one hand, a way to say "Here are your orders", on the other hand it was born out of necessity... Royal Navy was not an egalitarian service, so I believe this form was adopted as an option to order around people who would be subordinate to the person issuing order as an officer of the Royal Navy, but may very well be superior in terms of social position, without saying "you are hereby ordered".
The use of the word "required" (in various written forms) is well documented back to the Middle English, so it is not hard to make the claim that the form "requested and required" in naval orders was the left over from really old times, from various legal documents and correspondence.
Legal documents frequently use this kind of double-barrelled terminology; e.g., heirs and assigns, cease and desist. They are called legal doublets.
Adding to the above, because historically, something may be an order, but its also expected an officer will want to do the right thing anyway, so theres no need to be hard headed about it.
So its a polite request to hand over command, as well as a requirement as an order. In a way, its the same reason your boss at work says "can you do X today?" not "I order you to do X today". They could just order harshly, after all, they are your boss and you're their employee. But can you see how, even despite that, the softer version feels nicer and works better?
Command is a tool. Its always there. But you dont need to be a dick with it, and use it unnecessarily harshly, with experienced willing subordinates. The soft skills matter a lot more.
And so, like his military predecessors, Picard is cordially asked to hand over command using a tried and accepted formula, that recognises he will (very probably) wish to do the right things once he knows whats wanted by his commanding officers, not just the equivalent of "Here are your orders! Obey, drone!"
Picard is required because a valid order has been issued to him by a person vested with relevant authority. But he is requested because the Captain of a ship is, in many senses, its absolute master, and his authority is not so easily brushed aside - even if, in the world off his ship, he answers to others. At least on a symbolic level, he cannot merely be ordered around on board of his own command.
As @AcePL notes this is a well-known anarchronism from wet Earth-navy times.