Would the Anglican Church consider themselves Protestant?

The existing answers approach the question of whether Anglicanism is Protestant by considering the position of Anglicanism and deciding whether it is Protestant. My view is @curiousdannii does this best. However, I want to approach the question from another direction. It is the case that people have varying, and often imprecise, definitions of what Protestant means. There is an expression "the Queen's English" meaning the use of the English language in the way the Queen uses it.

Two days after the death of George VI our present Queen made a declaration promising to respect the Protestant nature of the Church of Scotland. This however is not Anglican.

At the first opening of the UK Parlaimentafter her accession she declared that she herself is a faithful Protestant.

More significantly at her Coronation on June 2nd, 1953, the following exchange took place:

Archbishop. Will you solemnly promise and swear to govern the Peoples of the United Kingdom of Great Britain and Northern Ireland, Canada, Australia, New Zealand, the Union of South Africa, Pakistan, and Ceylon, and of your Possessions and the other Territories to any of them belonging or pertaining, according to their respective laws and customs?

Queen. I solemnly promise so to do.

Archbishop. Will you to your power cause Law and Justice, in Mercy, to be executed in all your judgements?

Queen. I will.

Archbishop. Will you to the utmost of your power maintain the Laws of God and the true profession of the Gospel? Will you to the utmost of your power maintain in the United Kingdom the Protestant Reformed Religion established by law? Will you maintain and preserve inviolably the settlement of the Church of England, and the doctrine, worship, discipline, and government thereof, as by law established in England? And will you preserve unto the Bishops and Clergy of England, and to the Churches there committed to their charge, all such rights and privileges, as by law do or shall appertain to them or any of them?

Queen. All this I promise to do.

The "Protestant Reformed religion" is a reference to both the Church of England and the Church of Scotland.

So, yes, the Church of England is Protestant

The physicists come up with a particular definition of "colour" and tell us white is not one. Those who say the Church of England is not Protestant do something similar, but according to the official meaning of Protestant, yes Anglicanism is Protestant.

It's simply a feature of a human language that a word has many connotations and the one connotation that is meant in an oral/written communication is the one in the speaker/writer's mind, to be deciphered by the listener/reader. So in your conversation with them, they were probably referring to a specific Protestant aspect/connotation pertinent to the rubric of your discussion, that Anglicans do not have.

One possible example of that connotation is the tight association of the word "Protestantism" with the personalities of Martin Luther and John Calvin in informal parlance; in a word association game, "Luther" would most likely come up first given the clue "Protestant". Since neither personalities were directly influencing the Anglican church politics during its formative period, if your conversation partners have in their mind those 2 personalities as the foremost indicators of "Protestantism", it makes sense for theologically uninformed people that they would deny that Anglicans are "Protestants", although Anglican theology was influenced heavily over the first few hundred years by Reformed theology.

To address the question in your 2nd paragraph about Anglican's self identity, I'm offering a more foolproof/bottomline aspect that is mentioned only obliquely by curiousdannii and davidlol's answers, that an Anglican church is Protestant because it is NOT in communion with the Catholic Bishop of Rome and does NOT recognize the authority of his curia, mainly the Magisterium (which primarily is invested in the CDF).

This severance was done from the very beginning (see my other answer), thus even before the adoption of the Thirty-nine Articles of Religion, and most likely before the term "Protestant" gained widespread use in the English language in England. Therefore, Anglicans are Protestants only retroactively. Historically they may not self-consciously identify themselves as Protestants, since in their mind it's more the state-based separation from Rome (the creation of the legal entity of the Church of England as independent from Rome) rather than all other connotations that other Protestant churches used to self-identify themselves with (like association with the personalities, sola fide, sola scriptura, etc.)

How about Anglican churches that in the past century or so have separated from the Church of England? Why they still don't prominently self-identify as Protestant, but as Anglican, even though they are no longer in communion with the Archbishop of Canterbury? My guess is that the reason is most likely connotation again, although the most prominent connotation of Anglicanism in the 21st century has to do with the culture of the way they "do church": the use of the Book of Common Prayer, their liturgies, their understanding of sacraments, their specific prayers and customs, etc. The brand "Anglican" fits their way of "doing church" better than the brand "Protestant" (as understood today).

The Anglican denomination is thoroughly Protestant; yes there is an Anglo-Catholic sub-branch, and some High Anglican liturgical features may have the trappings of Catholicism, but the core and heart of Anglicanism is thoroughly Protestant. Indeed Anglicanism is not only Protestant, but firmly within the Reformed (Calvinist) side of Protestantism.

Let's start with the defining document of Anglican theology, the 39 Articles. The Articles that express Protestant theology and practice include:

Article 6: which declares the apocrypha/deuterocanon to be non-canonical, and non defining for doctrine.

Article 10: which teaches the Reformed view of free-will, that humans cannot by their own nature strength bring themselves to turn in faith towards God.

Article 11: which teaches justification by faith alone.

Articles 12-13: which says that our good works cannot put away sin, nor do they make us receive grace from God.

Article 14: which rejects the Catholic teaching of supererogation.

Article 17: which teaches the Reformed view of predestination and election.

Article 19: which says that the Roman church has erred in matters of both ceremony and faith.

Articles 20-21: which teach that the Church and the church councils have subordinate authority to the scriptures.

Article 22: rejects purgatory, icons, relics, and the invocation of saints.

Article 24: which says that it is repugnant to pray or minister the sacraments to the congregation in a language they don't understand, contra Roman practice of Latin liturgies.

Article 25: which limits the sacraments to only baptism and communion.

Article 28: which says transubstantiation is repugnant and contrary to scripture, as well as that the elements of communion are not to be gazed upon, lifted up, or carried around.

Article 30: which says both elements of communion are to be given to all congregants, against Catholic practice which normally only gives the laity the bread.

Article 31: which rejects any idea of priests offering Christ again.

Article 32: which allows clergy to marry.

Article 37: which says the Pope has no jurisdiction in England.

And beyond the 39 Articles, the Book of Common Prayer is also thoroughly Protestant, and although it is not an accepted or binding document for Anglicans, the Westminster Confession of Faith (and other Westminster documents), the defining English language Reformed Confession of Faith, was primarily written by Anglicans. I see no reasonable way to exclude Anglicanism from Protestantism.

No doubt you'll find some 'Anglicans' who disagree with much of the above; there have even been some high profile atheist Anglican priests and bishops. But those renegades do not define what it means to be Anglican so long as the 39 Articles remain the doctrinal standard for the denomination. And even today the numerical weight is behind conservative Anglicans, not theologically liberal Anglicans, just maybe not in England or the USA.