Is it possible to guess the purpose of my potential PhD supervisor inviting me to lunch?
No one can really know what the exact motive is of the professor. However, if he or she is willing to take time to meet with you (especially for lunch, which means the person is also investing money into your meeting), I imagine that the program is able and willing to provide you with support and he or she is assessing your fit to work together. 

Be aware that this could be something of an interview where he or she is assessing a few applicants to see which one is best suited for working on their project. Therefore, I urge that you to prepare for the lunch as you would any other type of academic interview (e.g. review some recent work of this faculty member, prepare questions ahead of time for him or her). When I applied to my PhD program years ago, one faculty member invited me to meet with her, but ultimately chose another applicant to work on her project. I received support from another faculty member to work on a different project, but the invitation to meet by the initial faculty member did not indicate a guarantee. Best of luck!