Can Non-Linear Grammars generate Regular Language?
You are confused a bit I believe by using the terms grammar and language somewhat interchangeably.

A grammar is a set of production rules. A language is a set of words. A language is described, or generated by a grammar.

They are not the same and you should be careful at all times to see which is which.

There are a lot of terms such as "context-free", "linear", "regular", etc, that are direct properties of grammars. That is their original definition.

These terms when applied to a language, has a different, indirect meaning. When I say that a language L is X for those above properties, I mean that there exists a grammar with property X that generates L. 

Now this can get confusing. Let's take your original grammar. It itself is not linear or regular, but the language it generates is both. Why? Because I can construct another grammar for it which generates the same language, which has those properties:

$$S \to A$$
$$A \to aaA \mid B$$
$$B \to bB \mid \epsilon$$

If you want to be even stricter I could make it a strictly right-linear grammar without epsilon productions (other than the whole language being nullable):

$$S \to aA_2 \mid bB \mid \epsilon$$
$$A_1 \to a A_2$$
$$A_2 \to aA_1 \mid aB \mid a$$
$$B \to bB \mid b$$

Finally, we have that for language classification we have (among others):

$$\text{Left-linear} = \text{Right-linear}   =\text{Regular} \in \text{Linear} \in \text{Context-Free}$$


  Can I then say that above language is an example of non-linear, regular language?


That is indeed wrong, because the language above is linear. You are confused because you are looking at the grammar, which isn't linear. Just because a certain grammar for a language isn't property X doesn't mean that there doesn't exist another grammar that generates the same language that has property X.

That's why it's generally hard to prove that a language doesn't have some property X. You need to show that every possible grammar you can construct for that language doesn't have that property, because a single one is enough to give that property to the language.