Maimonides rules regarding a doubtful firstborn that we do not take it from him, but regarding a doubtful redemption of a firstborn donkey (the lamb with which they redeemed the donkey), if I remember correctly his position is that we do take it from him. And the distinction is that regarding the donkey, the Israelite has the presumption of prior possession, whereas the doubtful firstborn was born together with the doubt, and there was never definite ownership of it by the Israelite.

Maimonides’ position on this is an ocean, and I’m not immersed in it at the moment. See Kuntres Ha-Sefeikot and Takfo Kohen.

I looked again and saw that this is discussed in Rabbi Shmuel’s lectures on Bava Metzia 21a, section 4 — the laws of doubtful cases and possession regarding lost property — and the law of the presumption of prior possession is explained there.

The question appears in what he wrote in section 3:
And as for Maimonides’ basic position, it requires clarification, in light of what is stated in the responsa of the Rashba (section 311) regarding Maimonides’ view: that the rule that if a priest seizes it we do not take it from him applies specifically to a doubtful firstborn, “where the Israelite was not established in it any more than the priest,” since their possession of the fetus does not render them possessed of the newborn (and this reasoning needs further explanation, but this is not the place). Therefore, in the case of a priest seizing it, we do not take it from him. But in a doubtful redemption of a firstborn donkey, where the lamb was certainly established in the hands of the original owner, Maimonides agrees that seizure is ineffective (and see there, where he reconciles the Talmudic passage according to Maimonides). In truth, Rashba’s precise intent is itself not clear to me, and this is not the place; in any case, the later authorities understood him to mean this (see Even HaEzel on chapter 15 of the Laws of Robbery and Lost Property, law 1, and elsewhere): that a doubtful firstborn has no presumption of prior possession, and therefore if a priest seizes it we do not take it from him, but where there is a presumption of prior possession, as in the doubtful redemption of a firstborn donkey, he agrees that we do take it from him. According to this, Maimonides’ statement here is difficult (according to the Vilna Gaon’s explanation cited above): why does the finder acquire these doubtful items? Maimonides himself agrees that where there is a presumption of prior possession, seizure is ineffective and we take it from him, so we should follow the prior possessor, namely the owner. This is also the question raised in Even HaEzel there.

Now I thought of answering it another way: the finder of the lost item has a migo claim that turns him into the possessor both with respect to prior possession and with respect to ordinary physical possession. The claim is that he didn’t find the lost item at all; it had always been his.

1. When there are identifying marks, there is no migo, since it is clear that originally it belonged to the one who lost it.
2. If he is not in possession, migo does not help extract property, except according to the view of Nachmanides. And if he is in possession, no migo is needed. Migo operates in situations where the possession needs support, because there is some evidence against the possessor, or some weakness in his claim.
3. It may be that this is a migo without an actual claim, since he is not really claiming factually that it is his. His claim is a halakhic one: that he found it only after the owner had despaired of recovering it.
In other words: migo is meant to prove that I am not lying, because if I were lying I would have said something else. But here there is no suspicion that he is lying. The question is whether he is right or not. For that, migo does not help. True, migo as the power of a claim might perhaps help even in such a case, but in my humble opinion it does not.

1. Even if there are identifying marks, the finder can claim that he bought the lost item from the loser yesterday. So the fact that the loser recognizes the identifying marks of the object is not proof that it is his.
2. Does “migo to extract property” also apply against someone who is only in the position of prior possessor? Maybe when we say that we do not invoke migo to extract property, that is only against someone who is actually in possession — the object is in his hand — and not against someone who only has the presumption of prior possession over the object.
3. I meant more migo as the power of a claim. You can formulate my reasoning a bit differently: since I could have made some claim that would have caused me to win the lost item in court, then already now I am considered the one in possession of the lost item, even before I make that claim, and in a case of doubt — for example, doubt whether it was placed there intentionally or a legal doubt — the burden of proof is on the other side. This is true so long as the other side is not also actually in possession of the lost item, but at most only has the presumption of prior possession, which is a weaker presumption. But if he is actually in possession, then we do not invoke migo to extract property.

1. The claim that he bought it is weaker, because he admits that the loser was the prior possessor. I didn’t write that when there are identifying marks that is proof that the loser is the owner, only that it makes him the one in possession. A clear practical difference: if the loser brings witnesses that he lost the object, he will certainly remain the one in possession despite the fact that the finder now physically holds it. According to your approach, in such a case the finder would have to return it to him. I don’t think the halakhic decisors wrote any such thing. Especially since the discussion in the Talmudic passage is where the finder admits that he found it and that it is a lost item, and the entire discussion and doubt exist only because it is doubtful whether it was intentionally placed there.

2. When there is no actual possession, the prior possessor is considered the one in possession, and as far as I recall, quite simply migo does not extract from him.

3. Even migo as the power of a claim usually serves as evidence. It does operate through a mechanism of creating possession-status for the one with the migo, as we explained in the lectures on Petach Tikva in the name of Kehillot Yaakov, but its use is in the laws of evidence. I don’t recall a case where the dispute between the parties is halakhic rather than factual — meaning, the issue is not suspicion of lying — and migo decides it.

Regarding point 2, today I came across Ketzot, section 82 subsection 13, who writes that migo is effective to extract from the presumption of prior possession.

That is a novel idea and not agreed upon at all. Beyond that, he explains that this is because it does not leave from under his hand. But here he also actually holds it, so it seems that even according to Ketzot HaChoshen, migo would not help.

If he actually holds it, then he is truly in possession. Here we are talking about someone who is only the prior possessor, but not actually holding it. The side claiming migo — the finder of the lost item — is the one actually holding it.

My mistake — I’d already lost the thread. But the finder is not claiming that it is his; he is claiming that it is not the loser’s, and therefore he acquired it. I think that is different from a dispute with a prior possessor over having bought it from him. Especially since here the finder is basically claiming “perhaps.”

Following up on this, today I came across the Ran, from whom it seems that the owner of the lost item — the loser — is not considered in possession of it. Bava Metzia 24b, s.v. “Does it say ‘these are his’? It says ‘he is not obligated to announce.’” Explanation: since the majority are non-Jews, the obligation of returning lost property does not apply, for we follow the majority here as in other prohibitions. Nevertheless, it does not become his either, because since it is possible that a Jew will come and provide its identifying marks, he should leave it. And this is not because we do not follow the majority in monetary matters, for we only say that where money is established in the possession of one of them, in which case we do not remove it from its possessor because of a majority… Rather, here it is only by rabbinic enactment that they instituted that we do not follow the majority in this matter, since it is possible that a Jew will come and provide its identifying marks.
Still, it seems that most commentators follow Rashi’s approach, from which it appears that the owner of the lost item is considered in possession of it, but Ran’s understanding seems more logical to me.