Alexandria was a very important see in the early Church. It was a major center for Christian thought, with the first seminary. It was the first see where the bishop couldn't handle everything, he had to ordain bishops under him to help him. At the time the title for a bishop was "Father" so this bishop became a Father or Fathers, or in Greek, Papa Abba, or in English, Pope. This was long before the Church of Rome became large enough to develop this kind of hierarchy, but when they did they decided that only they should be called Pope. Communion had already been broken at that time.
Pope Shenouda has no authority over the other Churches making up the Oriental Orthodox Communion, so he does not have power like that of the Pope of Rome. (We also do not believe him to be infallible).
I'm not that familiar with the set up in the Anglican Church, but if his is just a primacy of honour, no, it's more than that for us. All the bishops and one archpriest make up the Holy Synod, which is headed by Pope Shenouda. He has no authority to proclaim new doctrines, but he does govern the Church. He has authority, and is to be obeyed by all the bishops, priests, and faithful. But there are procedures in place for removing a Pope if they are found unfit. That was done once not to long ago, and in that case three bishops are selected to rule until the Pope departs at which time a new Pope can be elected.
Pope Shenouda is the 117th Pope in an unbroken succession from St. Mark, who we count as our first Pope.