Death by many Magic Missiles

Let's start with when to make Death Saving Throws (PHB, 197)

If you take any damage while you have 0 hit points, you suffer a death saving throw failure.

So any time something gives you damage while at 0 hit points, you suffer a saving throw failure.

As quoted above in the question, the language for Magic Missile is separate per missile. Each missile strikes simultaneously, but the damage from each is separate. If we are separating damage, we are separating those Saving Throw failures.

The Concentration function is similar in that it triggers off of damage (PHB, 203):

Whenever you take damage while you are concentrating on a spell, you must make a Constitution saving throw.

However, it also goes on to state:

If you take damage from multiple sources, such as an arrow and a dragon’s breath, you make a separate saving throw for each source of damage.

Jeremy Crawford specifically ruled on Concentration saves and Magic Missile requiring each missile to force a new save.

Roll for each missile

The real question

That leaves us the real question here as to whether or not Magic Missiles are separate sources of damage, or if the missiles hitting concurrently represent a single taking of damage.

Given that Crawford seems to believe that magic missiles are multiple sources that require a roll per source, it seems you can extrapolate that the missiles are also giving damage separately (while hitting concurrently) and thus forcing death save failures for each magic missile. The concurrent nature of the strike doesn't override the multiple deliveries of damage from separate sources.

Caveat Emptor

If you are a DM and planning to target a PC like this, you must beware of hard feelings at the table. Typically death saves are a PC thing, and if you are going to remove that by going after unconscious players then the possibility of hurt feelings is real. Most monsters are more concerned about taking out other creatures who are alive rather than double-tapping any downed creatures. If there was a real reason by the NPC to do this, then that's different - but if you're just going after unconscious players because they're easy targets to completely kill you may suffer backlash from your table.

No, Magic Missile does not hit the target. It sails right through it. Magic Missile targets a creature, and in your mind the illusion is a creature.

Each dart hits a creature of your choice that you can see within range.

The end result is that the Magic Missile is cast, the slot is expended, but it does not hit anything unless you specified more than one creature and any of those other creatures are valid targets (non-illusionary creatures). This may give the caster reason to believe his target is not quite what he expected, and may give cause to inspect the Silent Image more closely.

As for what actually happens in the narrative when this occurs?

Let's say the caster casts magic missile and decides to hit one (unknown to him) illusionary target and two other real targets, one missile each. The missiles streak towards their targets, and hit the two real ones. The third target is an illusion and can't be hit, so the missile passes right through him. This may tip the caster off that this may be an illusion. This is the most likely scenario.

The other option is for the DM to say, "that one isn't a valid target. Try again." This, of course, tips the player off, and lets the player choose a new target for his missile. This way involves metagame information, pulls the players out of the scene, and is probably a lot less fun.