If you are considering a project the whole process, part of the process, or when running a process and you are managing it then you are a project manager. Then other side is what definition of project and project manager you take (from what organization) to measure it. Saving Changes...
It's a temporary endeavour undertaken to create a unique product, service or result. A project is temporary in that it has a defined beginning and end in time, and therefore defined scope and resources.
So, what you are doing comes under project banner for sure Saving Changes...
The first or second time you hire someone it might be considered a project, but even then if your firm has an established process for recruiting it would be an operational process.
Just because something has a start/end, consumes resources and possesses some uncertainty doesn't mean we should treat it as a project as the overhead costs of project management might not be justified.
Purely technically, I guess every job requisition you fill *is* an individual project (it's temporary, it provides a service, there is a defined end result in filling a position).
But is that project being managed? That is the question to ask. What are the steps taken against each of the knowledge areas? In each of the process groups?
I would suggest looking up the matrix of these elements on this site and measuring those aspects against everything you do when conducting your recruitment. See how much applies.
(NB: There have been cases where specific recruitment assignments - especially for niche skills - can been treated as projects. For fun examples, see Akira Kurosawa's Seven Samurai or the derivatives of this excellent film.)