Scholars, you have enough friends. You don’t need your mentor to be your new pal. Mentors, you don’t need to invite your mentee to your upcoming dinner party. It’s okay to be friendly. It would be strange if there wasn't a bond developed through your mentoring relationship - but boundaries are important.
If you get too involved in each other’s personal lives, lines will blur. Friendships can also make it more difficult for mentors to offer unflinching, constructive criticism. You might get along swimmingly, but it’s best to table the friendship and keep things professional for the duration of the mentoring relationship. You can hang out all you like once the mentorship is over. Remember, your mentoring relationship is about career and personal development, not making a new friend.