I know precisely what a "third-party library" is, so I assume that the "third-party" in this case simply is an arbitrary person or company other than the developer?
Does it have to do with "first person", "second person", and "third person" grammatical points of view?
The fact that there is a "third party" suggests that there is a "first party" and a "second party" as well. Are those terms well defined?
(I'm not a native english speaker.)
If you are developing software for a client, then there is a contract between you/your company, and the client/their company. These are the two parties to the contract. Anyone else, not bound by the contract, is a third party. It's used wherever a contract exists between two parties to mean anyone not bound by the contract.
There is no fixed meaning to which of the two parties is 'first' and which 'second', usually you will think you're the first party, and the client the second, whereas the client will think they are the first party and you the second, in a similar fashion to first (I, we), second (you), and third person (he, she, it, they).