Identity productions in regular grammar
Yes the language is regular.

Productions 2 and 3 are like transitions in a finite state automaton: $B \to aC$ is like moving from state $B$ to state $C$ while reading $a$; $B\to \varepsilon$ indicates that $B$ is like a final state where we can stop the computation (i.e., end the dirivation in grammar terminology).

The production $A\to B$ is like an $\varepsilon$-transition in finite state automata. This may be practical, but does not change the language generating capabilities of the grammars.