Mentioning that you had another offer can be tricky. Did they ask you if you were seeing other companies? Did you just put it out there? Did you put them on a deadline?
Mentioning that you have another offer is seen as a negotiating tactic. Then they have to compete/bid against the other offer, and you are removing some of their leverage/power. A strong reaction to such a move can be to cut off the interview, so to regain back their power.
Yes, you're correct. But it's best to walk away from companies who try to own you to that level. If a good company respects and wants you, they are completely OK with giving up some of their leverage in order to gain your trust as well as they can.
I think they may have asked me, although, to be honest, I'm not 100% how it came up. It wasn't me putting them on a deadline though, I just try to be pretty open about things in general.
Mentioning that you have another offer is seen as a negotiating tactic. Then they have to compete/bid against the other offer, and you are removing some of their leverage/power. A strong reaction to such a move can be to cut off the interview, so to regain back their power.