If a Stirge chosen as a Beast Master ranger's animal companion has had the dragon's Breath spell cast on it, can it use its action to spew magical energy (per dragon's breath) while blood draining?
It's a bit unclear
The case for action
While monsters can take standard actions in addition to their specific action options listed in their stat block, the Blood Drain action does seem to specifically contain the draining once attached.
Blood Drain. 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.
The specific Stirge action of Blood Drain provides the melee weapon attack to attack, but then it later (in the block) it also states that (emphasis mine)
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.
My reasoning here is that because the draining is included inside the blood drain action, and that the use of it says instead, the implication is that the *blood drain** action is first an action melee weapon attack to attach, and then it is an action for the draining of blood.
Note that the action isn't Proboscis insertion with the result being an out of action drain, but the name of the action itself is Blood Drain.
Outside of the Blood Drain action, the stat block does provide another action option to release. This suggests the need to separate it out, which the draining does not do.
But there is a case for no action
The whole phrase of "at the start of the Stirge's turn" does seem to suggest that it just happens. And it also frees up the Stirge to release itself after it finishes draining rather than having to wait another round.
One thing to consider is that this does mean that a Stirge could Dodge and force disadvantage on attacks while it's feeding or it could even disengage prior to detaching to get away scot-free, which I'm not sure is the intent.
This DM leans towards action for a Monster
Because of direction inclusion of the draining within blood drain action and that it utilizes "instead of attack", I tend to lean towards this. It also limits the Stirge's ability to do this.
However, your use of this is as a an animal companion. Because of that, I'd tend to be more lenient, especially with the life expectancy of the Stirge due to its AC and HP. However, allowing it for the PC means I'd allow it for the monsters, too. Which the party may not enjoy.