Yes, according to the DMG.
The DMG (kind of) assumes that you will place requirements and limitations on character creation. For example, it suggests that you give a handout that includes (DMG 26)
Any restrictions or new options for character creation, such as new or prohibited races.
On the same page, DMs are told to be accommodating with their backstory requirements:
Even if you want all the characters to have grown up in the starting town, consider allowing a recent arrival or a transplant if the player's story is convincing enough. Suggest alterations to a character's story so it better fits your world, or weave the first threads of your campaign into that story.
So the direct question is yes, the system allows DMs to have some control over the players' backstories.
The main pro is cohesiveness
I have played and DM'ed in games where restrictions were imposed, and in ones that were not. The main benefit to DM control of backstory is that it allows the DM to be much more flexible in where the game starts. For example, having all the characters be part of a military or mercenary group opens up certain plot threads, and having all the characters have some ties to the main antagonist helps set up the campaign.
For characters, it really cuts down on time where the characters warily get to know each other (see: almost every superhero team-up), and allows the group to dive into the action. Whether or not this is a good thing depends on your personal taste.
The main con is regretting your restrictions
In one game we've played, the DM restricted the class and race options pretty significantly. He later mentioned that he regretted putting such strict restrictions on us. Likewise, you might find that you've painted yourself into a corner with the backstory that you've required.
Honestly, though, if you have player buy-in, I don't think there are any significant cons to this approach. As the DMG quote above states, you can probably shoehorn most of the characters your players want to play into your predetermined backstory.