Lo-Fi Games

The first line of dialog I saw when I first met crab raiders was "You're not going to survive long without a crab." So I went to their town, bought a crab and listened carefully to the crab care tutorial. Next I went to their armor shop to do some shopping. However the merchants called me "crabless" and didn't want to sell their stuff to me. I tried to prove I had a crab by dragging my crab into their tiny hut, I walked around the village with the crab on follow and disguised as one of them. They still wouldn't believe me. I thought maybe they remember me entering the village without a crab and not consider me true enough. So I did the same in the other village, except I already arrived there with my crab. On the way I was constantly attacked by crab raiders. The villagers didn't attack me but still called me crabless and didn't want to trade. How do I become a crab owner in their view?

that sounds more like a bug.

Being "Crabless" is actually what they refer to you when you're not their ally, it has nothing to do with Crab ownership. To become a "Crabbed One" you must either talk to their queen and join them that way, or win the Crab Tournament.