The Bible does not record angels having wings at any point, but this is a popular depiction of angels. Do any Christian groups teach that angels do have wings, in extra-biblical materials? Where does the idea come from?
The Bible actually does indicate that angels have wings, as both cherubim and seraphim are angels.
In the year that King Uzziah died I saw the Lord sitting upon a throne, high and lifted up; and the train[a] of his robe filled the temple. 2Above him stood the seraphim. Each had six wings: with two he covered his face, and with two he covered his feet, and with two he flew. 3And one called to another and said: "Holy, holy, holy is the LORD of hosts; the whole earth is full of his glory!"
Isaiah 6:1-3 ESV
The cherubim shall spread out their wings above, overshadowing the mercy seat with their wings, their faces one to another; toward the mercy seat shall the faces of the cherubim be.
Exodus 25:20 ESV
He put the cherubim in the innermost part of the house. And the wings of the cherubim were spread out so that a wing of one touched the one wall, and a wing of the other cherub touched the other wall; their other wings touched each other in the middle of the house.
1 Kings 6:27 ESV
It is also assumed that the four living creatures are angelic beings, since it's hard to imagine what else they could be:
And the four living creatures, each of them with six wings, are full of eyes all around and within, and day and night they never cease to say, "Holy, holy, holy, is the Lord God Almighty, who was and is and is to come!"
It would make sense angels have wings if you are to accept that cherubim and seraphim are indeed angels. I don't. You are hard pressed to find anything in Scripture that indicates they are. It just isn't there. C andS beings are not messengers, comforters or have any attributes of the other angels except to say they worship and obey God.
The common accounts of angels (not C or S) never mention angels having wings. As a matter of fact they were always mentioned as being who appeared as though they are human.
How much sense would it make for Paul to implore his readers to be kind to everyone because they never know when they are entertaining angels? If an angels is standing beside me with wings, I think I would have inkling there might be an angelic being in front of me.
As far as the statement angels are "floating" proves they have wings, then you have to assume to the same for Jesus after he ascended into heaven or wings were needed for Elijah.