The HP loss via Blood Drain is damage.
The HP loss due to the Blood Drain is indeed damage: this can be confirmed by the following question on the SAC:
If a shadow rolls a critical hit, does it reduce the target’s
Strength by 2d4, as well dealing the extra necrotic damage? No. A critical hit lets you roll damage dice twice. An
effect that deals damage is one that reduces the target’s hit
points. The shadow’s Strength reduction isn’t damage, because it has no effect on the target’s hit points.
A further evidence is in the description of the Horned Devil's Tail attack:
Tail. Melee Weapon Attack: +10 to hit, reach 10 ft., one target. Hit: 10 (1d8 + 6) piercing damage. If the target is a creature other than an undead or a construct, it must succeed on a DC 17 Constitution saving throw or lose 10 (3d6) hit points at the start of each of its turns due to an infernal wound. Each time the devil hits the wounded target with this attack, the damage dealt by the wound increases by 10 (3d6). [...]
We can hence generalize to get the definition of damage:
Damage is HP's loss
Since the Stirge's Blood Drain reduces the HPs of a creature each turn the monster remains attached to the target, the 1d4+3 does count as damage.
There are several spells that can grant resistance/immunity to all damages.
Some spells provide protection from all kind of damages:
[...] While the target is within 60 feet of you, it gains a +1 bonus to AC and saving throws, and it has resistance to all damage.
The sphere is immune to all damage, and a creature or object inside can't be damaged by attacks or effects originating from outside, nor can a creature inside the sphere damage anything outside it.
You are immune to all damage until the spell ends.
It is reasonable to consider it of piercing type. Resistance/Immunity to piercing damage should work.
It is true that the damage dealt has no explicit type stated, and Darth Pseudonym in their answer provides some reasons for considering this damage of the necrotic type.
I nonetheless would rule that the damage due to blood loss is of the piercing type.
Let's parse again the statistics of the attack (emphasis mine):
Melee Weapon Attack: +5 to hit, reach 5 ft., one creature. Hit: 5 (1d4 + 3) piercing damage, and the stirge attaches to the target. While attached, the stirge doesn't attack. Instead, at the start of each of the stirge's turns, the target loses 5 (1d4 + 3) hit points due to blood loss.
and its further description
Blood Drain. A stirge attacks by landing on a victim, finding a vulnerable spot, and plunging its proboscis into the flesh while using its pincer legs to latch on to the victim. Once the stirge has sated itself, it detaches and flies off to digest its meal.
These two pieces of text tell that the the proboscis is the body part used to deal the initial damage and the subsequent 1d4+3 due to the blood loss. Since the on-hit damage is of the piercing type, it is not unreasonable to consider the following damage of the same type.
Furthermore, this interpretation agrees with the description of the piercing damage (emphasis mine):
Piercing. Puncturing and impaling attacks, including spears and monsters' bites, deal piercing damage.