I'm aware that Wizards can learn spells from other spellbooks and scrolls they find, as long as they take the time and money to learn it. If there is an enemy, such as a Mage that is listed in the Hoard of the Dragon Queen supplement, that is listed as a caster and said to have Wizard Spells, does this mean we can assume they are a Wizard class and they have to have a spellbook? Is there any chance a PC Wizard could learn the spells off this defeated enemy?
As you say, a Wizard can learn spells from spellbook they find. So the question here is whether wizardly enemies will have spellbooks on them when they are defeated.
That's entirely up to the DM. Like any piece of loot, any enemy could be carrying a spellbook for whatever reason. If the DM wants the players to find the defeated Mage's spellbook, they will. There are a multitude of reasons for why a Mage would or wouldn't be carrying their spellbook on their person - it's the DM's choice to give the players access to this (fairly valuable) piece of treasure or not.
Remember that a Wizard doesn't require their spellbook to cast their spells (unless they want to cast them as rituals), so it's perfectly reasonable for a Mage to have hidden their spellbook somewhere once they prepared their day's spells.
So if you're asking this as a DM, the answer is that it's up to you. If you're asking as a player, the answer is that it's up to the DM, but it might well be worth your while to search the lairs/houses/dens/homes/castles/towers/whatever of enemy Mages you defeat.
It's up to the DM, but it is dependent on which feature they use to cast spells whether or not they would use a spellbook
According to the Monster Manual introduction, if a monster has the Spellcasting feature then they have class levels:
A monster with the Spellcasting class feature has a spellcaster level and spell slots
The other ways that monsters can cast spells, or produce spell like effects are through the Innate Spellcasting or Psionics features described in the MM introduction.
If we take one of the spellcasters from the Lost Mine of Phandelver (LMoP)
as an example. Their stat block states:
Spellcasting. The mage is a 4th-level spellcaster that uses Intelligence as its spellcasting ability (spell save DC 13; +5 to hit with spell attacks). The mage knows the following spells from the wizard’s spell list: [...]
They use the wizard's spell list, and have class levels, and thus is a xth level wizard (see the spoiler for the exact level).
By virtue of them being a wizard, they require a spellbook to prepare their spells. They do not however require a spellbook to cast spells they prepared in the last long rest they had access to a spellbook.
The Monster Manual also says:
A stat block rarely refers to equipment, other than armor or weapons used by a monster.
You can equip monsters with additional gear and trinkets however you like, using the equipment chapter of the Players Handbook for inspiration and you decide how much of a monster's equipment is recoverable after the creature is slain and whether any of that equipment is still usable.
If a spellcasting monster needs material components to cast its spells, assume that it has the material components it needs to cast the spells in its stat block.
As a result we have two possible options:
- The designers didn't bother to put in the spellbooks because they didn't think it important to the adventure/assumed the DM would know to include this after having read the Monster Manual (in which case you should add a spellbook with appropriate spells.
- The wizards presented have lost their spellbooks and are only able to cast the spells that they last prepared until they replace their spellbook (in which case there is no spellbook to recover).
Were I to run this adventure as a DM, I would use option 1 and add in any additional spells as appropriate for the particular NPC. For example:
Glasstaff is brewing potions of invisibility, so it makes sense for them to have Invisibility in their spellbook.
I'd say it entirely depends on the NPC wizard, in question. If they are just "built for the moment", something for the party to fight, and probably kill, if you are doing well, and if he just needs access to the spells he'll cast, then no, he probably won't. If he's part of a group, or has a lair, he can, conceivably, prepare his spells, leave his precious book at home, and be good as gold.
My understanding of spell prep, in this edition, is that the wizard only needs his books to change his prepared spells, after a long rest. So long as he likes what he's got, he doesn't need the book; the next day, his mana recharges, and he's got the slots back. He still has the same prepped spelss as yesterday. If he needs something different, then he needs to heft the heavy tome, thumb through it, and recommit words, gestures, and such, to his memory. If he has a safe place to leave it, I'd leave it, rather than the cliche "party got captured, and lost all their loot." Especially in a new world where magic items are rare again, losing mundane gear is a hassle, but losing your only copy of your spellbook...no thanks.
If your NPC wizard is "important", such as a recurring villain, who harasses the party regularly, and improves as they do, I'd say it's a bit more likely he could have the book on him. If he's likely to do ritual casting, setting traps for you, or some such, he might even need to. Defeating this enemy could justify acquiring his books of spells, making it a worthy addition to your treasure pull, and let's be fair, it's what the wizards want (especially if scrolls are also rare). If he's "just a schlub", though, I'd say he probably doesn't have it, and you'd need to find the fort he came from, or tower he resided in. My "typical" War Wizard of Cormyr, for instance, is learned enough to be pretty sure what his ideal array for battlefield casting will be, and leave his tome safely in the barracks; if he doesn't die, and things change, he can get back to it.
As an aside, I wonder if there is any reason a wizard with a suitably strong familiar couldn't have it "hold" his spellbook(s), dismiss it into whatever pocket dimension it resides in, and then call it up to peruse his book, when convenient. Seems easier than dragging several heavy books around, or several other options. I might've just found another reason to like familiars. ;)