Is this a regular grammar? I went through a question asking me to categorize the following grammar.

$$S → AA, S → AB, A → a, A→BB, B → b, B → e$$

From the production rules, clearly it is Context-Free. But it accepts a finite set of strings.
$\{e, a, aa, ab, abb, ba, bba, b, bb, bbb, bbbb\}$
which is regular language.

So, is the above grammar regular? Though it does not follow from the rules.

Basically my question is: Is the grammar $\{S → AA, A → a\}$ regular?.
The answer is no. Whilst the language is finite and therefore regular the grammar itself is not regular. I think you already knew this as you said that this grammar's production rules are not of the form of a regular grammar. 

The point here is that just because a grammar isn't regular doesn't mean the language it generates must not be regular.