Edit OP clarified after I wrote the answer that she is a student, not an employee. Nonetheless, I will leave this answer here as I think it is still useful to others who might be in a similar situation. It could also apply to the OP's case if "manager" is taken to mean her Dean/Principal/Career Counsellor/whoever is her "supervisor" in regards to this interaction with the "mentor".
It does sound like he is misusing the mentoring to try to hit on you. Sending an email describing what to expect in the mentoring, getting to know coworkers better, and sharing one professional photograph is okay, but the rest of it is entirely unnecessary.
If you are uncomfortable with his advances (as I expect you would be), you have a few options to stop it.
- Give him a polite warning
Hi so-and-so, I would prefer to restrict our mentoring to the workplace, and not extend it to our personal lives outside work. Do you think that would work?
Ask your manager to assign you another mentor (self-explanatory, I would presume).
Report the situation to your manager.
Let your manager deal with the situation. They are better skilled in dealing with such situations.
Which of the options you would use depends on your situation and the company culture.
Be aware that the option 3 will create at least a black mark in his personnel file (which most "corporates" maintain per employee) if your claim is found to be valid. If the company doesn't usually take these complaints seriously (such companies exist!), then he might get the "license" to create further problems. Tread carefully if your company is one of these.
You also need to consider how much "political clout" he has in the company. While sexual harassment in most companies is serious business and would lead to severe disciplinary action, it is possible that some "investigators" would pretend to look the other way if he has "important contacts" or other kinds of importance in the company.
Edit In hindsight, and after reading Jane's comment about this point in her answer, I realised my laziness has led to a misinterpretation, which I will clarify.
If he turns out to have a political clout, it does not mean you should simply put up with the harassment. Rather, you should consider this factor if you choose to take the formal complaint route. In a company where sexual harassment cases are taken seriously, and if nothing else works, it is a no-brainer.
However, if the company is the type which will ignore complaints from lower-ranked employees to protect the high-ranked ones, then you have to be aware that you are in for a long-battle, which might involve taking the matter outside the company (that is, legally). We all have limited time on this planet, and whether solving this company's broken system is worth the sizeable amount of your time is for you to decide.
I am especially inclined to point this out because I get the impression from your description that all the "creepy" communication with him happened over the phone, for which there is no paper trail (unless you had the amazing foresight to record all his calls).End of edit
That said though, if he were not so creepy as you have described here, a case could have been made for this being a cultural difference (since I do not know anything about his culture). In the Eastern world, it is not that uncommon for people to try establishing personal relationships before professional ones. However, even by the standards of Eastern culture, asking for FaceTime unrelated to work and demanding sharing of all personal and family life is way over the top.