That's exaxtly how it is.A collar is similar to wearing a wedding ring, it is a symbol that you are taken, spoken for, owned property. The leash identifies the owner.
Putting a collar on immediately changes one's personality, you become docile, submissive it puts you in a different state of mind. A collar reminds you of your place and purpose. It is almost always used as a training device. It is sexy, exciting and instant arousal.
But also a way to show loyalty I think. He enjoys having us both in a collar mostly to confirm his control.