They are not synonyms. First, "compania" is not a word in Spanish. I assume the OP meant compañía, which does translate to company. Second, compañía usually has the sense of "company" used in phrases such as a theater company or a company of soldiers.
Yes, it can mean company as in business firm or organization. However, it's better reserved for entities like corporations or industry organizations. If you use "compañía" to refer to a little corner grocery, no one is going to stare blankly at you. They will certainly understand what you mean. Indeed SpanishDict offers a couple example sentences with just that kind of usage.
I would, however, point out this definition from RAE:
Sociedad o junta de varias personas unidas para un mismo fin, frecuentemente mercantil.
The rule of thumb is that countable nouns that immediately follow ser do not carry an indefinite article. There's another rule of thumb that says if it's usually assumed you are speaking of a single thing then you can omit the indefinite article, "I'm going to buy a car" - "Voy a comprar coche." Both of these "rules" would apply here.
Thus, if "una" was used in this case, it would be to emphasize that the speaker is talking about only one.