All firms appointed as a Market Maker on a US Options Exchange have an obligation to provide continuous quotes in a given 'class' of options (all of the options for a particular underlying stock). An exchange participant generally needs to apply to an exchange to be a market maker in a given class of options.
The rules and the amount of continuous quoting varies from one exchange to another, but in general all Market Makers are required to (i.e. 'must') provide continuous quotes in each appointed class at least 60% of the time and the series, and no more than $5 wide except in certain situations.
Many exchanges also have different classifications of market makers and special ones such as "Designated Primary Market Maker" (CBOE), "Primary Market Maker" (NASDAQ ISE), "Lead Market Maker" (NYSE Arca Options). I am not sure if any of them use the same "Designated Market Maker" terminology that NYSE uses.
These special market makers generally have higher quoting obligations (e.g. coverage 99% of the time and the series) and sometimes have additional responsibilities, such as deciding when the market in a particular options series should be opened or re-opened after a halt. However not all exchanges require these special types of market maker to list a class of options for trading.
The rules and obligations for a particular type of exchange member can be found in the exchange's rule book.
Historically an exchange was owned by its market makers (or 'specialists'), where each market maker owned a 'seat' (an actual chair) where (actual) people would have to go if they wanted to trade a particular security. As they were the counter-party for every order transacted on the exchange in a given security, they could make quite a lot of money.
This monopoly has been gradually chipped away over the years with most seat holders receiving a share of the stock of the exchanges that subsequently went public and the number of seats being increased resulting in more competitive markets. Now a 'seat' is simply a market maker permission a firm has to apply for which comes in different flavors depending on what advantages the market maker wants and is willing to pay for and obligations they are willing to accept. Most options exchanges now are fully electronic so the limit on the number of market makers is no longer constrained by available space on the trading floor.
Designated Market Makers for Stock Options?
Yes. ("Designation" is not required).