The Anglican Catholic Church is Anglican because it stems from the ancient Catholic Church in England. It does this through its belief in the traditional teachings of that original Anglican Communion . It holds to the Revelation of Christ, Scripture and the Seven Councils. It has and teaches the necessity of , the apostolic succession in Catholic Orders stemming from the Apostolic College and receives the Faith as taught by the fathers of the first thousand years. Lack of money enforces the use of the Anglican Missal which seems to me the greater part of the Prayer Book but, unfortunately this encloses some Roman additions which were part of the Ordo calendar and some enthusiasts use these, I don't because the Calendar contains sufficient to satisfy my Anglican needs and I wouldn't want to deprive the weaker bretheren anyhow!
I guess the church I found myself in and very happy with is the of the catholic variety(APCK). From my uneducated view and for all sack/ purposes utter confusion of all the divides/ arguments and divisions it is very proudly Anglican. The Mass has all the bells and smells like I expected from a catholic church(or for that matter eastern), but has the identity of Anglicanism. No one in the church I know of are Romans Catholic wanna be. A lot of things I consider specifically Roman Catholic I do not see here. The ultra high exaltation of Mary as a Co-redemmer, Purgatory, The pope. We do celebrate ST. Days according to the BCP and during Mass I do here something that sounds a lot like the Roman Canon with it's litany of saints. Again I am confused by a lot of labels and the alphabet soup of Anglican groups. Vaguely think I go to a high church, but I am not even clear on that. What brought me to Anglicanism was all I wanted was sacraments and scripture. So I sort of shun away from the politics.
Your story does sound like mine. I grew up in a high church variety that was never confused about its Anglican identity. Only when probing on social media do you actually find out that you fall under an Anglican "group". A clear example is that my Anglicanism is always viewed with suspicion because I was never thaught the 39 Articles and in practicality I do not see a place for them in the church that I was brought up in. Only recently has our national church started to re-brand itself into what I would call "protestant...we've never seen our church this way" and the locals are not very impressed and members are calling for the re-reinstatement of certain practices and devotions.
Anglicanism has always considered itself to be reformed Catholicism-that is, biblical Catholicism. The Church of England Institute of the Articles of Religion in order to correct some Roman, though not Catholic, practices that kept him to the church over time and to return us as a church to our biblical roots - our apostolic roots. For me, the Articles of Religion are just as important today as they ever were.