Just checking: clans can trial for specific giftakes or gentic material right, and hence the matrilineal progeny of those giftakes are hence eligible for the associated bloodname right?
Corollory: the clans do not declare trial of possession for specific bloodnames. Is this correct?
Outcomes:
1. It's possible for a bloodname to be shared by several clans but end up with all the holders from one clan.
2. A clan holding the genes to a bloodname may decide not to use them, effectively making the bloodname 'exclusive' again. Of course, they can change this at any time.
3. When a clan takes a warrior as abtakha (what's the etymology of this word anyway?), the genes are not conferred automatically and must be trialled for separately - we know this for a fact.
4. The clan which trialled successfully for the gentic materials (eg. warriors X, Y, and Z) is only limited to using those materials and their progeny correct? Likewise, the clan which lost these materials are no longer allowed to use them.
Yes to your first two questions.
1. In theory sure. But if that other Clan legitimately owns genetic legacies of that Bloodname, they can produce warriors too.
2. Effectively... yes. Though I think everyone would be hesitant to call the bloodname exclusive to one Clan. That language "Exclusive" implies something that would not technically be true in this case. I feel like they'd be more likely to say "The Jade Falcons share this bloodname with the Wolves, but the Wolves have chosen not to breed any sibkos." I say this because a Bloodname being Exclusive has been treated as something almost special, important. Akin to saying unit X has "never lost a battle." You don't want to muddy the picture with weird gray area stuff, and the Clans tend toward straight honesty in many ways. So IMO that's how they'd tend to comment on such a thing. A great example is Clan Smoke Jaguar, which absorbed all the Clan Mongoose bloodnames but refused to use them, so they were content to let those genetic legacies sit on the shelf forever. Several were already shared with other Clans or became shared with the Cloud Cobras in particular. But in the 3050s-60s those Mongoose-origin bloodnames did not appear as Exclusive to the Cobras or any other Clan. The ones we've been able to identify using the list of original 40 Mongoose warriors listed in Operation Klondike don't appear as exclusive to any Clan. Even though the Smoke Jaguars just refuse to use them. So I don't believe anyone calls such Bloodnames Exclusive, they remain shared Bloodnames.
3. We don't know the etymology of that word as far as I know. But you have the right idea about the warrior and the genetic legacy being two separate things and that ownership of one does not automatically convey ownership of the other. The warrior might change Clans via becoming a bondsman/bondswoman to another Clan. While the genetic legacy may be won in a Trial of Possession.
4. Yes, if a Clan wins the genetic legacy of Warrior Tom Yung, then the can use that genetic legacy to breed more warriors, using it as either the genemother or genefather. If the former, then those offspring are eligible for the Yung bloodname. If another Clan swept in and won all the genetic legacies of the Bloodnamed Yung warriors within a Clan in one or more (likely multiple Trials of Possession, then that Clan would lose access to the those genetic legacies and they'd effectively lose the ability to breed any more sibkos who would be eligible for that Bloodname.
Elaborating a little more on Clans refusing to use a genetic legacy. We have seen this in canon, but it's somewhat rare, rare enough that if it happens at any scale it's usually called out in the books. The Wolves were rather checkered in their approach to using the Widowmaker genetic legacies (using some but not others). The Smoke Jaguars refused to use the Mongoose legacies at all as an intentional slight against that Clan. There are probably other genetic legacies that have some form of taint on them. Like that warrior committed a crime or something like that, some action in life that soured everyone's opinion of that person. We've also seen attempts to block the use of Bloodnamed warriors' legacies who developed serious natural health problems while they were alive.
But otherwise, without a reason, the Clans tend to use as many genetic legacies as possible, because the gene pool is already so small. The exceptions are rare enough and serious enough that they tend to be called out like they are noteworthy important events and facts in the books. But if you are trying to craft a story or character where this is a thing, there's certainly room for it, we do have plenty of precedents.