Angels appear to have six limbs: arms, legs and wings. Whatever they are, then, they are probably not vertebrates, I.e., animals like us with a spinal cord enclosed in a backbone.
A superfluity of appendages
No living or fossil vertebrate has more than four limbs. Whales and snakes, among others, have abandoned some or all external limbs, but no known vertebrate has ever added any. Even fish, which have all kinds of external structures for locomotion and/or stabilization, are built around a basic 4-limb bauplan (body plan) of paired pectoral (shoulder) and pelvic (hip) fins.
It may be that four is the general optimum number of limbs-enough to get the job done; not so many as to squander energy in the developing fetus or in practical use. Likely it is a mostly incidental consequence of having evolved from an ancestor built around the 4-limb structure, which, in most vertebrates today, appears as four legs. Some groups have modified forelimbs into either arms (primates) or wings (bats, birds), but no species has both… or they wouldn’t have a leg to stand on.
As the pictures all show, however, angels appear to have perfectly competent arms as well as feathered wings emerging from their shoulder girdles. It’s hard to figure how a creature could have gotten there (angels) from here (the vertebrate 4-limb body plan).
Those wings won’t work
Evolution is essentially a bricoleur, in throwing up new species it makes do with what is already in its tool kit rather than designing from scratch. All the principal existing body plans date back to the Cambrian Explosion of 530 mya (million years ago). Arthropods produce the most limbs, ranging from approximately 14-limbed insects (bet you thought they only had six; several of their mouthparts are actually modified limbs) to many hundreds, even approaching 1000 on a very old millipede, but it’s not ‘hard’ for them to add limbs. They do it by multiplying body segments, each of which produces a pair of ‘limbs’: antennae, jaws, wings, legs.
Vertebrates are not segmented like arthropods, however, and besides, angel wings violate the vertebrate body plan in a more significant way than merely number-they apparently attach to the dorsal (back) side of the shoulder (pectoral) girdle. Check it out: all vertebrate arms, legs, wings, fins attach to the ventral (belly) side of the pectoral and pelvic girdles. Where would powerful flight muscles attach: how would they be stabilized. A vertebrate flapping wings like an angel’s would torque its spine into an unholy mess.
It’s one or the other
And a third thing! Depictions show angels with feathered wings but long silky hair. No living species forms both hair and feathers. Of course, Gabriel’s flowing locks might not really be hair, but the kind of long, soft, mostly unbarbed feathers called filoplumes, thought to play sensory and/or decorative roles in birds. Filoplumes may sense the position of contour (body) feathers so that they can be kept properly aligned, they may play a role in providing body awareness in flight, they may make birds more attractive to the opposite sex. If Raphael’s filoplumes were indeed misconstrued for tresses, one wonders how angels make use of them.
Of course, hair and feathers are not features that fossilize well, so it’s possible that somewhere out there, just waiting to be found (or perhaps already found and mistaken for a feathered dinosaur?) is a furry, fluffy multi-limbed fossil of the vertebrate/angel last common ancestor. One could probably better spend ones time searching for the Lost Ark of the Covenant. Oh, wait. That’s in some dusty U.S. government warehouse, isn’t it? Or, was it a church in Ethiopia?