This doesn't seem like a valid regular grammar; my instructor says it is
Your teacher is wrong, unless of course s/he does not adhere to the wikipedia definitions. It is "technically" equivalent to a regular grammar, but that is not a well defined concept. What can we allow as extensions that give us "technically" regular grammars. 

There is a concept that is perhaps suited. That is self-embedding. A variable $A$ is self-embedding if $A\Rightarrow^*uAv$ for non-empty $u,v$. A grammar is non self-embedding if none of its variables is self-embedding. Every non self-embedding context-free grammar has an equivalent regular grammar, which was already observed by Chomsky. With this concept we extend the regular grammars, while we are certain to stay within the regular languages.

And, since you are asking more details, your instructor is wrong again on the single nonterminal remark. Such grammars also include $S\to aSb\mid \varepsilon$ for which every sentential form has (at most) one nonterminal, but for which the language is non-regular. In fact these languages are called linear, and form a family between regular and context-free. 

Note that $S\to AB, A\to aA|\varepsilon, B\to Bb|\varepsilon$ defines a regular language (mixing leftmost and rightmost). On the other hand $A\to aB\mid\varepsilon, B\to Ab$ is nonregular. This exemplifies the non self-embedding distinction.