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anothername: The only other thing that pops my mind with your guidelines in the OP (and surprisingly not mentioned for some reason would be Morrowind (https://www.gog.com/game/the_elder_scrolls_iii_morrowind_goty_edition).
And here I was getting my hopes up that I'd get to be original by linking Morrowind here on GOG.

The game has a massive amount of area, you can play 3rd person view, active real time, should be no internet required after downloading the installer.
Take a look at Gothic 3(or even the whole series; although I've only played #3 so far so can only comment on it).

Like some of the other recommendations, it doesn't meet every one of your criteria, but it meets several.

Yes - Open world (large and seamlessly open - no loading zones anyplace, including going into towns and dungeons [both large and small])
Yes - character customization
Yes - fantasy setting
No - mounts!!!
Yes - loot (excellent loot system; is very rewarding to "explorers")
No - possibly a class with pets, like the warlock in WoW (BUT - you can turn into various animals, depending..)
Yes - third person view preferably, not isometric
Yes - I would prefer an action-oriented combat system, but one similar to WoW is fine. I would prefer not being a turn-based system (It isn't turn-based)

The world consists of 3 "biomes" (Temperant with mountains, plains, valleys), High mountains (snowy and icy; one of my favorite areas in any open-world game - it's so well done IMO), and Desert (hot and dry). All of the areas have ocean borders (yes you can swim; and there are places to swim to); and transiting from one biome to another is "seamless" (you just make your way between them; either overland on-foot or via the fast-travel system).

Exploring is rewarding in that there are tons of hidden nooks and crannies - usually with stuff to loot - including some really good loot. So climb those peaks and check behind those boulders <g>.

There are multiple "factions"; and you can earn friendship with them or turn against them.

Etc., etc.

It's a great open-world game IMO.
Sacred 2 comes to mind...