"No, I have a bother and sister" is grammatically acceptable as is "No, I have a brother and a sister." Those two sentences, however, have slightly different nuances in meaning.
The first example focuses on the two people as members of a single collection, namely the speaker's siblings. It denotes that the set of the speaker's siblings differ with respect to sex while subtly implying that the similarity of siblinghood outweighs the difference in sex. It would be a perfectly grammatical answer to the question of "Is it true that you have three siblings," a question in which sex is not of importance.
The second example focuses on the two people being similar in one respect but differing in another relevant respect. It would be a perfectly grammatical answer to "Is it true that you have two brothers," a question that implies sex is a relevant consideration.
Notice that I constructed these examples so that both nouns started with a consonant. The choice of whether one or multiple indefinite articles are needed does not depend on whether the nouns' initial phonemes are in the same or different classes. It depends on whether what is being stressed is similarity or difference.
In response to the comment by the OP, I think you misunderstood my answer.
Of course it is true that an indefinite article is either "a" or "an" depending on whether the initial letter of the immediately succeeding word is a consonant or a vowel.
But that rule applies to WHICH indefinite article to use, not how many.
HOW MANY indefinite articles to use depends on what you want to stress. In your example, if you want to stress that the service being discussed combines the power of an application and a data base, then just ONE article implies that, and the article to be used will depend on the immediately succeeding word. If what you want to stress is that one service provides two kinds of benefit, then use TWO articles.