There was no notice of another player building it. Is it possible that some player started with it? I'm playing a relatively casual game with just easy opponents though.

I tried again using a civic trait, 10% research bonus from both kingdom and sovereign, plus a first turn Inspiration on my starting town. Went straight for Trading as my first research and still no Bazaar in the build menu.

Just started a quick game and purposely did nothing but get to trading. For my first level up on city I even picked Fortress to make sure that the bazaar isn't a Town only building. Didn't build a merchant in that town even. As soon as I hit trading Merchantcross Bazaar is available to build in that city.

Another possible reason you can't build is lack of space possibly? Are you surrounded by swamp tiles in that city? If so, you'll need to raise land on some tiles to create some more room for city growth. If not that, I'm not sure. Oh by the way, this game I played with Tarth, didn't want to have monsters coming in and wrecking my quick game Is it possible certain races can't build this? I usually get this building built, but I play primarily with men and not fallen. Pariden, Altar, and Tarth are my most played, and I have definitely built it with all of them.

My understanding is that it's the first person to queue the Merchantcross Bazarre (or any unique building) that gets it -- unless they cancel of course.  Once it's placed in a queue, it's theirs.  It's sadly not like Civ where it's a race to build and you can win the race even if you start later by outproducing your opponent.  You'll probably see the notification in 20-30 turns when the AI finishes the building.

Plus the cities they have been built in: second the motion.

Recently I haven't noticed the AI going for wonders other than the Tower of Erog, and it should notify you when they get built.

I'm ok with 1st-to-queue, except a notification that someone else has started might be nice.  1st-to-finish might mean a lot of rage when you invest a lot of time in the building and another faction beats you by two turns.

Galactic Civilizations had a first to finish system.  And there was rage (on my part) when an AI opponent beat me to a galactic wonder, and all my production on it was wasted.  But it did let you build up your production capability so that you could start later and still finish it first.