In effect, yes. There is nothing wrong with saying "second party" because people generally know what you mean but there is no true weight or significance to it. Bloodborne was created and developed by From Software, so people call it "second party", but it's owned by Sony, the publisher, which makes it a first party work.
Same is true relative to the studio. It's either independent from the publisher or it's not, so it's either a first party studio or a third party studio.
Though, really, all these phrases are relative to the system we're talking about at the time and can be changed around depending on perspective, so it's not really a big deal. They're just representative terms. For example, in a classic business model, gamers (the customers) would actually be the "second party," the sellers (publisher) would be the "first party," and any outside sources (independent developers) would constitute the "third party."
Then what about the Rabbid characters created for Mario + Rabbids Kingdom Battle?
That would be nothing more than a 1st and 3rd party collaboration between two separate IPs that are owned between the two.
The problem with use of the term "second party" is everyone wants to use it as a definitive lable for most devs in question. However they will still use the 3rd party lable for bigger dev teams that are known to make multiplatform games, despite the project itself fitting the bill for a second party game.
From Software entered a second party contract with Sony for Bloodborne. Platinum entered a second party contract for Astral Chain. Game Freak entered a second party contract for Pokemon.
In each of these cases, the publisher/platform holder owns the IP and controls what happens with the franchise. Sometimes these are one off deals (like bloodborne and FS), and sometimes this leads to a long standing partnership between the two where they only focus on one playform (like Pokemon and GF).
Now a game like Persona 5 for example, despite being exclusive, is a 3rd party game as Sony did not publish the game nor do they own the IP. This also Applies to Bayonetta 2 and 3, where Nintendo published it and might have rights to the individual title, but owns nothing of the IP and has no control of the franchise's future.
There is really a lot that goes into it and there is a lot of ground to cover, but trying to learn it on forums is not the way to go as people let theur personal interests get in the way. This is why you get people that say Bloodborne is a 1st party game when it is not, just because they own the IP just to fluff 1st party output. Then you get people on the other side saying it is a 3rd party game to make it sound like Sony has nothing to do with it, which is even further from the truth.