Well if you’ve read the history of the Papacy, stuff happening like this isn’t strange. I think it’s good for the Pope to drop a title like that, since Christ is the head of the Church in heaven and on Earth. No Patriarch is a vicar of Christ or shadow of God on Earth. However just a question to any Roman Catholics here, how can you remain a Roman Catholic when dogma and theology is obviously constantly changing whenever a Pope feels like it?
This would be a great question to ask over at One Bread, One Body–the Catholic forum here on CF.

Not only would you be in a better place (so to speak), but you would also likely get many more answers. But, moderation-permitting, I'll try to answer a bit here. Be forewarned though that I won't go beyond my initial answer, lest this become too much of a tangent.
To begin, I would say that I'm actually a well-read student of the history of the papacy, if do I say so myself. It's one reason that I'm Catholic today (which may surprise you). The title "vicarius Christi" is actually quite unassuming when translated and understood contextually. It simply means that, like any bishop or successor to the Apostles, the Pope is a representative of Christ. It does not mean a replacement (God-forbid!) or anything like that. In Catholic theology moreover, because of the Petrine ministry uniquely conferred on St. Peter (and by extension, his special successors), the Pope of Rome has a unique representation of Christ in having an authority no other bishop has. Hence, it became customary in the West to refer almost exclusively to the Pope of Rome has "vicar of Christ" to emphasize this unique authority. This does not change however that every bishop and patriarch is also a "vicar" of Christ.
Additionally, Catholics do not believe that the Pope is the head of the Church per se. Christ is the head. The Pope is merely the "visible head" or "earthly head" who exercises authority given to Him by the true Head, Christ. A better way of understanding the Pope's role is not "head of the Church" but "prime minister" or "royal steward" (the latter being, we believe, very Biblical, cf. Matthew 16; Isaiah 22).
Lastly, your characterization of Catholic dogma as always evolving based on papal whim (sadly a common idea in recent centuries) is very skewered, at least from our perspective. We believe that it is true that Catholic dogma develops, but it does not change from one thing to another new thing. More could be said, but I'll note simply that your characterization is also not in my own experience.
When I was on the journey to becoming Catholic (I was raised Baptist), I felt that the easiest way to disprove Catholic doctrine would be to show that it had
fundamentally changed over the centuries. So I attempted to demonstrate just that to myself. I never found it; not in the way that it threatens what would become my chosen belief. I have never really ceased "sniffing out" such seeming contradictions actually, and I still haven't found a single one that represents a break with previous teaching or an entirely "new" doctrine.
In my personal experience, as both a lay Catholic and a student of history & theology, I simply do not see dogma changing based on papal whim. What I do see are careful developments in understanding the already established doctrines handed down by the Apostles (and ending with them; cf. Jude 3).
I hope that begins to answer your question, and if I have stepped over any bounds unwittingly, I apologize.