Anglicanism is somehow 'between' Catholicism and Protestantism.
To answer your first question, the phrase "via media" means "the middle road", and one of the commonest contexts where this phrase is used is to describe the Church of England as a "via media" between the Catholic Church and the various Protestant churches. The above-linked Wikipedia page for the phrase is almost entirely (barring the first paragraph) about Anglicanism, and the Oxford Dictionary's example sentence using the phrase is "the settlement has sometimes been described as a via media between Catholicism and Protestantism", which clearly refers to Anglicanism.
Your second question assumes that the Church of England is just another Protestant branch, but it's not really seen in that way. This is a complex issue, going back to the Elizabethan Religious Settlement in the 16th century and the Thirty-Nine Articles drawn up to describe the religious direction that the new Church of England intended to steer between Catholicism and Protestantism. I'm not a religious scholar, but there's a lot of information about this if you'd like to read further on it. Quoting from Wikipedia:
The settlement under Queen Elizabeth I (from 1558), known as the Elizabethan Settlement, tried to find a middle way between radical Protestantism and Roman Catholicism, the via media (a term that actually only became current in the 1620s), as the character of the Church of England, a church moderately Reformed in doctrine, as expressed in the Thirty-Nine Articles, but also emphasising continuity with the Catholic and Apostolic traditions of the Church Fathers.
I also found this interesting, as it goes directly against your interpretation of the Church of England as a branch of Protestantism:
Wishing to pursue Elizabeth's agenda of establishing a national church that would maintain the indigenous apostolic faith and incorporate some of the insights of Protestantism, the Articles were intended to incorporate a balance of theology and doctrine. This allowed them to appeal to the broadest domestic opinion, Catholic and otherwise. In this sense, the Articles are a revealing window into the ethos and character of Anglicanism, in particular in the way the document works to navigate a via media ("middle path") between the beliefs and practices of the Lutheran and of the Reformed churches, thus lending the Church of England a mainstream Reformed air. The "via media" was expressed so adroitly in the Articles that some Anglican scholars have labelled their content as an early example of the idea that the doctrine of Anglicanism is one of "Reformed Catholicism". [source]
One important difference between Anglicanism and most branches of Protestantism is apostolic succession: just like in the Catholic Church, Anglican bishops claim direct descent from the Apostles of Jesus, while most Protestant churches either don't have bishops at all or don't care about apostolic succession.
One important similarity between Anglicanism and Catholicism is in the leader of the church. The Supreme Governor of the Church of England is the UK monarch, just like the leader of the Catholic Church is the Pope. Both of these are internationally recognised and important figures, with a clear line of descent which stretches back for centuries. Compare them with the various Protestant religious leaders, who are not politically important or well-known outside of their own respective faiths.
The divisions and subdivisions within Christianity are a vast and extremely complicated topic. But on a general, simplistic level, it does make sense to consider three main branches of Christianity in England, with Anglicanism standing between Catholicism and Protestantism. This explains the allegory in A Tale of a Tub.