YES, the creature can attack the stirge (nasty little creatures).
The system is written to do what the ability or effect says just as written (Rules As Written) and it does not state that you cannot attack in the description of the stirge's Blood Drain ability. There is also no general rule covering attacking the stirge using this ability.
However there is the general rule regarding situational advantage and disadvantage that could come into play should you, the DM, decide they it is warranted:
Advantage and disadvantage DMG p.239
Advantage and disadvantage are among the most useful tools in your DM's toolbox. They reflect temporary circumstances that might affect the chances of a character succeeding or failing at a task.
Consider imposing disadvantage when:
• Circumstances hinder success in some way.
• Some aspect of the environment makes success less
likely (assuming that aspect doesn't already impose a
penalty to the roll being made).
Given this, as a DM I myself would impose disadvantage for this situation, possibly unless the attacking character uses a finesse weapon and dex based attack. A character in this situation may be able to gain advantage on the attack to cancel the disadvantage. Not only is this an appropriate situational modifier, in my opinion, but perhaps more importantly it adds a bit of flavour and drama to the proceedings.
There is a closely related question: "Does a character have a penalty attacking an attached Stirge?
Weasels, in real life, lock their jaws when they bite down. That's what this is modeling - it bites, and now its jaws are locked and it's attached to you. It doesn't have to make any additional checks to stay attached, grapple or otherwise. It is "grapple-like" in that it's attached to you and loses its Dex bonus, but other things (grappling/grappled conditions, etc.) don't apply.
You can move, but it doesn't unattach the weasel. If it's just a normal sized weasel, it's probably pretty easy to move with it attached to you. If it's giant, then it's probably not so easy; you would need to do a grapple check yourself and use the "move" option. It is not technically grappling you, so they don't say you are grappled. It doesn't move into your square (unless it was a Tiny one and had to in order to attack you).
The way to unattach the jaws is use grappling yourself to achieve a pin (and unlock the jaws). Or kill it, though technically in the real world this doesn't get the weasel's jaws unlocked, so it would be entertaining to make them go through more work to get it off them.
I had a weasel familiar in one campaign but using it to kill downed opponents was about all it was safe to do; because when it bites it's only doing like 1 hp/round and can easily be beaten to death by most any opponent with its newly lowered AC.
In e.g. AD&D 2e the weasels have more description that makes it clear that they are biting down and continuing to suck blood. This means that it's clear the weasel isn't supposed to be able to attach to multiple opponents. As this question isn't tagged [rules-as-written] I see no reason to go past common sense to "well it doesn't SAY it can't do that" lengths.
If you would like to make this easier, the Pathfinder giant/dire weasel just has "grab" and then the blood drain, so it works more within the rules.
Best Answer
No, there's no penalty to attack the attached stirge. If there was, it would describe it. I believe you're missing a few points, in any case:
First and foremost: The players have no idea what the stirge's capabilities are. Unless your players have read the Monster Manual, they won't know that the stirge has low AC and hardly any hit points. All they know is that a monster attached itself to them - they're going to want to remove it.
To a lesser extent: While attached, the stirge is guaranteed to deal damage. As a CR 1/8 creature, the stirge is only meant to be a threat to low level, probably first level, players. And at first level, where misses are common, guaranteed damage is a big deal.
Finally, the stirge's AC isn't that low. For a CR 1/8 creature, 14 is on the high side of AC. And a miss will probably cost a first level player half their health, so attacking is a risk. Manually removing the stirge is guaranteed to work, even if it doesn't guarantee that the stirge won't reattach.