No, in French you can use both "une compagnie" or "une société", as well as "une entreprise", "une firm", "une corporation" and "un business" or even "le biz" are all pretty common.
But of course, none of them are fully interchangeable in all contexts. You will typically not expect to hear "salut la compagnie" in a formal meeting with "les gens de la bonne société."
If you like synonyms, CRISPO gives 77 for société and 33 for compagnie.