Is this a monthly sub game?
No. It was originally, but that was changed just before the console release. Probably because monthly sub games don't really work on consoles because you already have to pay for online services in the first place.
The game is now "Buy once, play forever". Or until the server is turned off. Whichever comes first, naturally.
With that said, ESO does still retain a subscription service; however, it is now entirely optional. It is called ESO+.
What ESO+ offers is a choice in how you get your access to DLC. You can either not subscribe, and buy DLC the old-fashioned way, which gives your account permanent access to the DLC you choose to buy. Or you can subscribe. About £9 a month but you can get discounts by subscribing for multi-month packages. I subscribe to a three-month plan myself. £24 every three months is not much to spend on gaming all things considered, especially considering how much I now play.
With your subscription, you are given access to all DLC, excluding whatever the most recent "chapter", read, big DLC, is. Right now that's Elsweyr. If you were to buy the game today, and subscribe, you would have access to all DLC except for Elsweyr thanks to ESO+. You would have to buy Elsweyr to access that content though, until the next chapter, when it will likely be reclassed as DLC, making it eligible for ESO+.
As well as DLC access, your subscription gives you access to a number of perks. The free Crowns alone already equate to just over 100% ROI. Your £9 sub gets you 1650 Crowns a month. To buy Crowns alone, it costs £11 to get only 1500. I'm sure you can see the maths here. You also get double bank space, double furniture limit, 10% more experience and gold gain, a craft material bag with UNLIMITED space, and access to Crown Store discounts not available to non-sub' players.
If you're a time-hardcore player like me, it's a fantastic deal. But, ESO is perfectly playable without ESO+, and I know many top players who don't subscribe.
EDIT: Quite often ZOS runs ESO+ trial events, where all accounts are given a limited version of ESO+. Free accounts don't get the Crowns or Store discounts. However during these trials, all other ESO+ perks are available to everyone. Non-subs' are encouraged to grab otherwise locked skill lines during these events, and perhaps farm materials, as they will have the craft bag unlocked. What's more, after the trial, anything put in the craft bag stays there: You can take stuff out, but you can't put anything in.