Can I make a virtual printer driver for macOS?
As a service, it's gonna be working on the background and when any user pressed cmd+p to print out any document, mine app will be appearing in the list of printer on the default print out page of Apple.
Yes, it could be done, and you don't need a driverkit here.
You could find the code example here: https://sourceforge.net/p/pdfwriterformac/code/HEAD/tree/
This project does exeactly what you need, it's written in C and uses CUPS API.
Worth to mention, it is not running like a background service itself, it is instead creates some settings and binary which is used by cupsd system daemon, but this is how it works on macos.
I also think it is possible with some efforts to do the same with Swift, since you added the relevant tag to the question.