Is cobalamin and vitamin B12 the same thing?

Yes. Vitamin B12 refers to a specific group of cobalt-containing corrinoids with biological activity in humans. This group of corrinoids is also referred to as cobalamins. The main cobalamins in humans and animals are adenosylCbl, methylCbl, and hydroxoCbl. Food Cbl is hydroxoCbl. Cyanocobalamin is a synthetic form of vitamin B12 found only in supplements. Learn more here.