The best mentors do not set out to be mentors or see themselves as (primarily) mentors. Yet they push you, mentor you and won't let you go. Because of this it is not so easy to get someone to 'be your mentor' in a formal sense, or, if that situation arises, where someone has agreed to be your mentor, then it is not necessarily the real deal.
As others have said the mentorship arrangement only transpires over time, and on reflection. However it is something that you can find within days of working with someone, possibly even within minutes.
I believe some people are more prone to finding mentors than others. Think of school and the vast overwhelming majority of pupils that the teachers really could not care a great deal about. Then there are the chosen few, singled out for special treatment. A 'C' might be good enough for the normal demands of the course, however, for those 'mentor worthy few' it is not quite like that. Even if they do not care whether they pass of fail, for whatever special reason it is, teachers will not let them be like that. Consequently they get mentored whether they like it or not.
It is the same in the workplace, there are a lot of also-ran's. Someone who has to just pick up the phone and be in on time is not likely to be pushed to excel or develop. Others don't get to be left alone like that. This is a different pressure to 'must work harder and be more productive', more care is involved, whether desired or not. I don't think even intellect has much bearing on the deal, you don't have to be gifted for someone to pick you out for mentoring and there are plenty of very smart, intelligent people that just do not pick up mentors.
Outside of academia and learning for the sake of learning, one great pressure for 'successful mentoring situations' is where time is money and the job just has to be done for a paying customer. If you can help with that you can find yourself a most useful mentor along the way. Also, if there is a strategic need for some knowledge to be shared then that can help. If you are the only one willing to learn all the systems and do weekend call outs when things go wrong, there is a lot of incentive for others to get you fully up to speed.
I should also say that in mentoring situations, cock-ups are allowed. You can ask for help on a particular thing or break a particular thing three times and three times only. A fourth time and you are renegading on the deal.