One Christmas Eve many years ago, I attended a party in the early evening before going to church for the midnight service. When I told another party-goer this, he huffily said that he didn’t believe in anything the church taught because it is impossible for human-shaped angels to have bird-shaped wings because of musculature and the size of the breast bone. Random social conversations often flummox me, this one more than most. I can’t remember what I said or did in reply. Probably excused myself very soon after.
The only references to heavenly creatures having wings come in visions (Isaiah, Ezekiel, John are probably the best known), and those are never called angels, and none of the creatures called angels which interact with humans on earth are described as having wings. Isaiah calls them seraphim and only describes them as having faces, feet and six wings which operate in three pairs independently. Ezekiel calls them “living creatures … Their form was that of a man”, but they otherwise had four faces, four wings and various other obviously non-human features. John also refers to “living creatures” with six wings, one of which had “a face like a man”. Clearly, earthly laws of biology and physics do not apply to visions of heaven.
On the other hand, most humans who are visited by angels don’t even realise that they are angels until either the angels say so or the humans figure it out afterwards. Sunday’s reading at my church was the annunciation. Mary was “greatly troubled” not by Gabriel’s appearance (in either sense) but at the greeting which followed. Heavenly creatures are depicted as flying but angels on earth only appear and depart (with a few synonyms for each). The writer of the letter to the Hebrews urges his readers to “entertain” (or “extend hospitality to”) strangers, because “some have entertained angels unawares”. Clearly, angels are not instantly identifiable.
The Hebrew word מַלְאַךְ malakh* and the Greek word ἄγγελος anggelos both simply mean messenger, and in other contexts the word refers to undoubtedly human messengers. The phrases מַלְאַךְ יְהוָה malakh YHWH, a/the messenger of Yahweh and (ὁ) ἄγγελος Κυρίου anggelos kyriou an/the angel of the Lord are also used. Translators into Latin and English usually use nuntius or legatus to refer to human messengers and angelus and angel to refer to the heavenly ones. (* Compare the minor prophet Malachi, which may be his name or may simply mean “a messenger”.)
I doubt whether any part of the Christian church has ever officially taught that angels are exactly human shaped and have exactly bird-shaped wings. The fact the church has authorised or allowed depictions of angels with wings doesn’t mean that’s how they look. Jesus wasn’t born in the stable of a public inn, the magoi were not necessarily three in number and certainly weren’t kings, and the adult Jesus didn’t look Italian, or Flemish, or English.
If you simply don’t believe in angels or any other part of the Christian faith, then this is simply not a problem. I think that man latched onto the most convenient excuse without attempting to engage with the issues fully. I doubt that he’ll read this now, though.
I found a lot more information on TV Tropes, of all places, which includes biblical angels and others among literature and pop culture depictions.