Yeah do have to own stuff. Some of the factions are free though, obviously the first 4 are if you own War1(Empire, Greenskins, Vampire Counts and Dwarfs) however they each have their own DLC that add new units and legendary lords(Grim and Grave for empire/vc, warlord and king for greenskins/dwarfs). They also added Bretonnia as a free faction to Warhammer 1, so you get 5 factions with just War1. There were also time limited free factions from preordering War1(Chaos Warriors) and from preordering War2(Norsca). These are paid now but if you did preorder(or buy the games 10days after the release) you own them too. They also added some free legendary lords to the initial 4 armies. To unlock all the free stuff that wasn't in the base game however you might have to install War1, pick up the DLC on steam to add them to your library, then uninstall War1. I think you can unlock them with War2 installed instead but not entirely sure.
List of free content to grab:
Bretonnia(full army with 3 legendary lords)
Isabella Von Carstein(legendary lord for vampire counts, was added for free separately while Vlad Von Carstein was added directly into the game)
Wurrzag(legendary lord for greenskins)
Grombrindal the White Dwarf(legendary lord for dwarfs)
Jade and Grey Wizards(just some different lores of magic wizards for the empire)
As noted, owning the DLC or not doesn't affect wether the factions are present in the mortal empires campaign, only wether or not you can pick them. As long as you own both War2 and War1, you get 9 races(4 in war1, 4 in war2 and Bretonnia) but to get the others you have to buy their specific DLCs. It's possible they'll add a free faction to war2 eventually too. Other than that you can wait for sales for DLC, they generally get discounter by about half, leading to fairly good value for DLC(7-10euros for a faction but assuming you do only a single campaign, that's still a good 50hours worth, some of them have replayability with different enough lords like wood elves or alternative campaigns too). The only DLC that are kinda eh are the Legendary lords ones since they only add a couple new units and one lord per faction(2factions per DLC), but they're also cheaper, often hitting 4-5euros in sales.
Also Chaos Warriors and Beastmen are considered kinda lacking as factions, so would probably not recommend buying them unless you really want to play them. Chaos Warriors will also probably look like hot garbage once Warhammer3 releases and they have Demons of Chaos instead.