Actually, in a sentence like this, articles are required and we do use them. If you want to say that a person has a noun (a thing), you need to say "Jane has a dog (or hat, car, flower...)". If you want to say that the person has a specific one, you can use "the" instead of "a".
There are exceptions to this rule, of course (every rule has exceptions in English). If the noun is a mass noun like milk (you can't say "one milk, two milks, three milks..."), you don't use an indefinite article (a or an), so you say "Jane has milk". You use "the" to talk about specific milk, though, so you can say "Jane has the milk". You also don't use an article with a proper noun, so you say "Jane has Facebook", not "Jane has a Facebook" or "Jane has the Facebook".
If you don't like "a," make everything that uses "a" plural. We don't have a plural version.
She doesn't have a cat. - She doesn't have cats.
I don't have a phone. - I don't have [any] phones.
Sometimes this is a bit weird/incorrect though:
We don't have/own a house. - We don't have/own [any] houses.
He has a car. - He has cars.
This probably works best with negative sentences because changing it to plural is much less likely to change the meaning. Maybe this is why you don't hear "a" sometimes and it shows you that plural/singular matters.