Nevada: DMV
“D” is for “Department”.
In New Jersey the state DOT and the MVC have nothing to do with each other. DOT handles roads and transportation. The MVC handles licensing and registration. Neither is in the others chain of command.
In Michigan’s case, the Secretary of State office handles more than just stuff related to motor vehicles. It also has voter registration, elections, UCC, etc. within its purview.
It depends on what you are trying to do. For driver’s licensing, we have the South Dakota Department of Public Safety and for titling and registration and license plates and such we have the Motor Vehicles Division of the South Dakota Department of Revenue.
Here in PR due to the use of Spanish the place in different towns we go to get licenses and plates, the Driver Services Center, is called “the CESCO.” This is under a division of our Transportation Department, the Driver Services Directorate, whose Spanish acronym is DISCO and nobody ever uses it.
Are you saying that DISCO is dead?
NSW, Australia: It used to be the RTA (Roads and Traffic Authority), and now it is the RMS (Roads and Maritime Services), but to get a new licence you go to Service NSW, which handles customer-facing interactions for a bunch of departments.
In Georgia, the USA one, it’s called DDS, Department of Driver Services. What’s kind of amazing is all my experiences with them since becoming DDS have been pretty efficient and painless. I’m sure they will fix all that soon enough. 
UK: DVLA, the Driver & Vehicle Licensing Agency.
In New York, we stuck with the basics on this one: Department of Motor Vehicles.
I believe this reflects an administrative issue. A Department is generally the highest level of a government agency. A Division is an agency which is subordinate to a Department.
So in some states, the Department of Motor Vehicles is a separate agency. In other states, the Division of Motor Vehicles is a part of the Department of Transportation or the Department of Public Safety.
Victoria, Australia, we have the pretty aussie slangish sounding VicRoads.
Washington State is efficient and painless too. One commonality between Georgia and Washington is that the offices seem to be small outposts in convenient locations. Here I updated my registration in a store in the mall. In Georgia I think it was a kiosk in a Kroger. As opposed to the more stereotypical California DMV model of a soul sucking central office.
I know in Washington they use private agents at least for the vehicle registration offices. I’m not sure about driving licenses. That’s actually a separate thing. It may be that they’ve set up the agent system well enough for meaningful competition between offices.
. . . and everybody knows the secretary;s name, because it is prominently emblazoned on the sigh above the door.
In Western Australia, we have the Department of Transport, Driver & Vehicle Services. Colloquially, usually just “the Department of Transport”.
What would sound less vaguely internet-ish is something like “Vicco”.
Oh, God, I suck; I had to look this up. I’d been out of the country for a long time, and so the last one I remember was Terry Lynn Land, although I always remember Richard H. Austin for some reason. I guess he had some longevity. SecState for a state doesn’t have the same cachet as the federal level, I guess.
SAAQ - Société de l’assurance automobile du Québec (which would translate to Automobile Insurance Corporation of Québec). Which sounds an awful lot like an insurance company, but is a government body that insures vehicle-related injuries under the no-fault system, and also happens to be in charge of licenses.
Here is it AAPI, or AAPD (Auto accident personal injury / Auto accident property damage). The first time I heard AAPD I thought it was auto accident possible death, but was relieved to hear that I was mistaken.
Anyway it is the DMV in NY
Maine: Bureau of Motor Vehicles
New Hampshire: Division of Motor Vehicles