The feign death target will appear magical
Detect magic says
you sense the presence of magic within 30 feet of you. If you sense magic in this way, you can use your action to see a faint aura around any visible creature or object in the area that bears magic, and you learn its school of magic, if any.
Detect magic would not directly tell you that the body is alive and under the effects of a feign death spell. Rather, it will only tell you that the body is under the effect of a necromancy spell or other magical effect of the school of necromancy. Detect magic works on both objects and creatures. Furthermore, the clause in feign death does not make any provision for concealing itself, only the status of the body. Because the information provided by detect magic is non-specific and does not directly tell you about whether the body is dead or alive, detect magic would function normally.
But what conclusions can be drawn when you detect necromancy magic on a body? That the body is under feign death is definitely one possibility. A greater possibility is that the body is an undead creature of some kind, or soon to be undead\$^1\$. It could also be under gentle repose, although gentle repose also requires a copper piece laid on each eye so that would be a clue. It could also be magic jar or astral projection, although the body is catatonic rather than dead in those cases so proper inspection would reveal the body to not be dead. Those are all the necromancy spells from D&DBeyond which you could find on a body. There could also be other magical effects, such as from special abilities or magic items or homebrew.
To summarise, if you detect necromancy on an apparently-dead body, the body can be alive (as per feign death), dead (as per gentle repose) or undead (as per danse macabre or assumptions about magic\$^1\$). Since all three are possible, detect magic does not definitively tell you the body is not dead, so feign death should not inhibit detect magic.
If you suspect that you need to fool detect magic, there are a few things you could do. You could place some copper coins over the target's eyes to make them appear to be under the effects of gentle repose - which definitely applies to corpses. Or you can use the False Aura option of Nystul's Magic Aura to hide the magic.
Weakness to detect magic aside, feign death has other uses. It can be used to suspend the effects of poison and disease. And not everyone has detect magic available or always active, so feign death is far from useless.
- The typical spells for creating undead, animate dead and create undead, have a duration of instantaneous. This means that, very strictly speaking, a creature created by animate dead should not show as magical under detect magic. It is still possible to get zombies which detect as magical using danse macabre, because it has a duration of 1 hour. Also, even if animate dead should not strictly trigger detect magic, most adventurers and NPCs would likely assume that a corpse radiating necromancy magic must be undead. Further, in real play, I have found it quite common that undead will detect as having an aura of necromancy regardless of minutiae in the definitions of what is 'magical'.