They Are A Pair of Monsters With A Long History Together

You can appeal to Dungeon and Dragons tradition for a thematically (historically?) consistent relationship between an Otyugh and an Ettin in the manner you describe. The rules you are working with don't explicitly say one way or the other, as KRyan pointed out in his answer.

Citation: Article by Ed Greenwood in Dragon Magazine #96, pages 20-22, published in April of 1985, "The Ecology of the Gulgrutha"(1)

The Gulguthra, or Dung-Eaters(otyugh and neo-otyugh), are strange and ravening monsters, deadly creatures indeed. I asked Elminster about them some time ago ... (-- snip-- long article -- near the end we find)

• The otyugh is most often solitary, but may exist in symbiosis with another (often more dangerous or energetic) creature, such as a doppleganger, ettin, will-o-the-wisp, or even a beholder. For such creatures they serve to guard treasure, which they always conceal at the very bottom of their offal pile, hidden from view beneath the otyugh itself. Encountering an otyugh is bad enough but if you do see one, be sure to look around for another even more fearsome foe!

Ed Greenwood did the original writings on the Forgotten Realms D&D Fantasy setting. Whether or not that makes his article "canon" is beyond the scope of this question and answer. Even if the relationship isn't canon per se for D&D 3.5, it matches the "game history" of both monsters since their appearance in the First Edition AD&D Monster Manual as amplified by Ed Greenwood's articles in Dragon.

It appears that these two monsters arrived intact for D&D editions 3.0 and 3.5. (No massive changes). A lot of the monsters have remained pretty much intact from their original introductions.

If you are interested, color/flavor material for these monsters can be found in "The Ecology of the Ettin" (Dragon Magazine Issue #92) and "The Ecology of the Gulgrutha" (Dragon Magazine Issue #96), respectively.

(1) The "Ecology of _______" articles published in Dragon Magazine ran for a number of years. They added color and back story to the terse entries in the Monster Manuals for a great many monsters.

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.