This is a popular question on AskHistorians! Here's my take from awhile back, fleshed out with some details from this discussion and some new stuff because why not:

The Bible bequeaths two basic traditions of angels and what they look like: "basically men" (e.g. Mark 16:5, Gen 18), and grotesqueries with wings. As Ezekiel 1 describes the cherubim, "In appearance their form was human, but each of them had four faces and four wings... Each of the four had the face of a human being, and on the right side each had the face of a lion, and on the left the face of an ox; each also had the face of an eagle." The seraphim of Isaiah 6 have six wings and are covered in eyes, even under their wings!

The acid trip angel type pretty clearly comes out of ancient Near Eastern traditions--both the etymological roots of cherub and seraph and artistic depictions track back to Phoenician, Assyrian, Hittite art. (The Jewish Encyclopedia offers some nice illustrations.)

Strictly speaking, in the Hebrew Bible the cherubim and seraphim are not "angels" as such--angels were more strictly limited to "messengers of God" whereas the C&S are kind of "flyer-praisers of God." However, almost from the beginning of serious Christian theology, the category "angel" has been flattened out into generic non-human celestial creature. (That early date probably means it has roots in apocalyptic Judaism). Late antique and medieval Christian theologians used, as one method, a form of biblical interpretation called allegory, in which Bible stories, teachings, people, &c. actually stand in for a second, deeper meaning (relating to Christ, to morality, or to eschatology and the Last Things). Origen, Eusebius, & Co. all have different allegorical interpretations of the seraphim in Isaiah--but they all agree that the creatures are prophets--hence, angels. Furthermore, the theologians tag specifically the wings of the seraphim as signifying divine communication.

As for the human appearance angels, in the earliest Christian art they do not have wings! It is only after the conversion of Constantine and the full incorporation of Christianity into the Roman Empire that we have traces of that tradition. Scholars pretty much agree that Nike/Victoria is the iconographic or artistic model for humanform angels' acquisition of wings (Keck, Angels and Angelology in the Middle Ages does a good rundown of the bibliography on this, going back to 1898!). This parallels a contemporary adoption of other Greco-Roman artistic traditions into Christian art. It seems likely that the wings of the seraphim and cherubim, now understood as angels, influenced the choice of Nike as a model.

This medieval painting, of St. Francis of Assisi receiving the stigmata, shows how neatly artists could link the six-winged seraph with a more winged-human, less eldritch image. Given this particular scene, the seraph's somewhat flame-like appearance is also no accident. Not only are angels traditionally associated with fire and light, but the scene in the Legenda major of Francis being depicted is stocked full with fire imagery.

Later developments will see the addition of childlike angels (today's "cherubs") and eunuch (standing in for unsexed) characteristics to further distinguish angel and human in art /u/caffarelli has a post talking about eunuchs and angels, too, if you're interested!

Art was the "book of the illiterate," and make no mistake, it held and holds probably the most powerful sway over our conception of angels. Let me tell you about this theologian not named Dionysius. Yes, we know so very little absolutely nothing about him, not even his real name (in Acts of the Apostles, a judge named Dionysius and a woman named Damaris convert to Christianity on the steps of the Areopagus; our theologian adopts the name to give themself greater authority by picking a male biblical figure). And yet, Pseudo-Dionysius, as they are known today, manages to be this MASSIVELY important player in the development of Christian theology and, well, language: they literally invent the word "hierarchy."

More specifically, they invent the word hierarchy to describe first of all the ranking orders of angels! Incidentally, not only does Pseuds have the seraphim and cherubim as angels in the generic sense (there are also Angels, proper), but they are the highest two orders of angel. Relevant to present purposes, however, Pseuds spends a good bit of On the Celestial Hierarchy talking about all the different appearances angels are said to take:

Chapter 15: What are the formed images of the angelic powers? The fiery property, the human likeness, the eyes, the nostrils, the ears, mouth, sense of touch, eyelids, eyebrows, fingers, teeth, shoulders, elbows and hands, heart, chest, back, feet, wings, nakedness, garments, the bright clothing, the clerical clothing, the belts, the sceptres, the spears, axes, plumb lines, winds, clouds, brass, electron, choirs, clapping, colors of different stones, the likeness of the lion, of the ox, of the eagle, the horses, the differences in the horses' colors, the rivers, chariots, wheels

(trans. Colm Luibheid)

I remind you: this is the theologian who (a) invented the idea of a hierarchy of angels (b) invented the word hierarchy to describe them (c) was influential enough that we know Angels, Archangels, Cherubim, Seraphim, etc are all forms of "angel" (d) was influential enough that we use the word hierarchy at all. And they think angels can have the formed images according to all those appearances. And say so in this same text!

...But you don't think of angels as different colored horses and as wearing priests' vestments and as freaking wheels, do you? No, you think of Hallmark cherubs, or of the pity part they give all the other girls in Sunday school Nativity plays.

Follow-up question/addition as this made me curious of what we in Islam believe regarding angels.

In Islam the angel Gabriel is described as having 'thousands of wings' and that his face is so bright he appears to be literally made of light. In addition, the Quran refutes the widespread belief among pagan Arabs that angels were the 'daughters of god' and that angels are not defined by gender.

Are there particular reasons/origins for these kinds of interpretations? Considering how nomadic Arabs of the time had less of a connection to ancient cultures of the Near East (contrary to the regions where Judaism and Christianity emerged).

I think u/sunagainstgold has addressed most of the question; since you ask speciifcally about Zoroastrian influences I can go on a bit about that. This is one of those areas where many Biblical scholars are more comfortable with acknowledging Zoroastrian influence (the cynic in me suspects that the fact that angels are not that significant to modern theology plays a role in this). However, I believe this mostly concerns named angels.

In Zoroastrianism, worship is centered on the sacred heptad of Ahura Mazda and the six amesha spenta (roughly, Holy Immortal Ones). The amesha spenta embody six qualities: The power of rulership (Khshathra), Truth/Righteousness (Asha), Good Purpose (Vohu Manah), Health & Immortality (Haurvatat & Ameretat), and Devotion (Armaiti). The heptad is finished with the creator, Ahura Mazda, the Lord Wisdom, embodying that greatest of all human qualities. The Immortal Ones are furthermore associated with other yazata (divinity, worshipped being) embodying elements, and the picture is further complicated with the addition of Mithra (Covenant) and Apam Napat (Son of the Waters), embodying Fire and Water but also justice and social order - Apam Napat is since the work of Boyce identified with Varuna (Oath) of the Vedic tradition.

In addition, the concept of Fravashi, entities that protect righteous humans and maintain the order of the world, appears early on in Zoroastrian tradition (not in the Gathas, but in the Yasna Haptangaita, usually ascribed to Zoroaster's disciples). Depiction (based off Persepolis reliefs). It is one of the few Zoroastrian entities we actually have clear depictions of. The Fravashi is today best known as a symbol of Iranian nationalism, but they go very far back indeed. As you can see, the Fravashi was pictured as a man resembling a priest or perhaps a nobleman, with three pairs of wings, totalling six. This is in my opinion maybe a plausible source of the particular notion of guardian angels as men with feathery wings, especially given the presence of six wings on Seraphim* (pairs of triplets and vice versa are a typical feature of indo-European motifs). It is interesting to note that this depiction, rather than featuring the "ageless beauty and purity" of the young man, depicts a powerful man of high social status, perhaps imagined as the spiritual equivalent of high-ranking clerics or feudal lords who protected their subjects.

The presence of bird wings and tail feathers is easy to understand in a Zoroastrian context as birds - especially raptors and scavengers - are (per the Bundahishn) considered holy and benevolent creatures for their consumption of insects and pests and the devouring of the flesh of the deceased. Perhaps the Fravashi was thought to follow righteous humans, hunting down evil spirits in their presence like a bird of prey.

*The catch here is that we then have to accept Isaiah 6 as a later insertion - this is maybe plausible given the complex history of Ezekiel and the tripartite work that Isaiah itself is, but it's a bit out of my depth to speculate further on that. It is interesting to note though that the literal meaning of Seraphim appears to be "burning ones" - given that fire, Atar, is one of the most prominent benevolent divinities in Zoroastrianism.

With all this said, as /u/sunagainstgold notes, the depictions of winged chimeric creatures has a long tradition in the Near East, so if there is any connection between the bird-winged guardian spirit men of Zoroastrianism and Judaeo-Christian tradition, it's not a connection that one could ever prove to a very convincing degree.