Dreyfus has (correctly) pointed that Heidegger often treats Dasein as referring to individual human existence. But I think this ambiguity is part of the problem, since it is less clear that he is always thinking of an individual. In any case, the crucial question is about what it would imply if Heidegger were to say Dasein is a subject, an individual, a particular entity. This would seem to make of it something merely present-at-hand (occurent, extant), which is of course exactly what Heidegger thinks the tradition has misinterpreted. But we surely cannot dispose of subjectvity just like that and use Dasein without any rigor to mean anything one wants; culture in some places, individuals in others, and at some points not even that. One wouldn't gain anything from such gross conceptual simplification.
But I think the answer Heidegger wants to give is that Dasein is human existence in general- and that in this framework it is never the case that Dasein is first and foremost a subject without a world. Dasein is the entity which in any case anyone could call his own. Only entities such as Dasein can ask, assert and therefore relate to being (a dog cannot ask 'what is x?' or assert 'x is y').This is different from saying that all ontologies must begin from the mind and then proceed to constitute the world, since all we are saying is that ONTICALLY Dasein is the entity which everyone can call one's own. There might be occasions, however, when explaining equipment for instance, that an analytic of Dasein shows Dasein is not 'itself':
Perhaps when Dasein addresses itself in the way which it is closest to itself, it always says 'I am this entity', and in the long run says this loudest when it is 'not' this entity. Dasein is in each case mine, and this is its constitutio; but what if this should be the very reason why, proximally and for the most part, Dasein is not itself? [Ibid]Since the being of humans is characterized by this relation to being(existence), one can call oneself Dasein ontically as the being for whom being is an issue. This doesn't entail that when trying to explain the world we must begin by positing private mental contents opposed to external, objective stuff. It also doesn't entail that reality is 'made up' of ideas, or that what is first given is the 'I' of pure consciousness. In Heidegger's conception ofthe world, dualism is not tenable, simply because Dasein's existance doesn't begin by dividing subject and object, but by everyday involement, and dealings with Zuhanden entities (available, or ready-at-hand), which are not like this at all.
Heidegger is not saying that we are not entities or subjects but some vaguely unified spirit, but that characterizing Dasein in terms of subjectivity is to assume the world only gets experienced for and from a subject, that it pressuposes human existence entails an objectifying need for constant self-reference, whether tacit or explicit, in which the pure 'I' of consciousness accompanies all intentional acts. But 'In clarifying Being-and-the-world we have shown that a bare subject without a world never 'is' proximally, nor is it ever given. And so in the end an isolated "I" without Others is just as far from being proximally given." [B&T, Pg 152, 116]
According to Heidegger, the idea of a subject as that which weilds private contents, opposed to the world, blurs ontology up, since we have to now explain how this world comes into relation to being for a consciousness.This is either impossible (Kant) or simply unecessary (Husserl). But Heidegger thinks this problem arises from assuming that we live inside a world of ideas or phenomenal representations; the world is already disclosed for Dasein. Dasein is nothing but a relation to beings on the basis of the 'openness' of a world. There is nothing 'behind appearances' simply because what appears can only do so on the basis of the prior disclosure and assumption of the world. To deny the world is solipsism; to self-contain it isidealism. Dasein is nothing but being-in-the-world, since it has always grown into it without bulding it from theory.
Dasein is an entity, the entity for whom being is an issue. But this doesn't mean Dasein is different from the world, since strictly speaking Dasein IS being-in-the-world, and as such a world can only be for an entity like Dasein. (This is not to say the world is an invention of a mind, or that every would disappear or be destroyed if there is no subject, but that all relatedness to being can only occur for a being such as Dasein, with the possibility of calling into question 'that it is'). Dasein is not primordially an 'I' since the world precedes any such characterization. In its everydayness it doesn't deal with objects, but with equipment. It doesn't determine itself as a subject- a subsistent 'I' which and the other as an object, to ontologically attempt this reduction is what Husserl attempted and failed to do. In this sense, Dasein is 'mine' only insofar as I can claim my existence as belonging to myself as an entity, but this is no ontological determination.
That Dasein is not exclusively nor primordially a subject or 'I' does not exclude that Heidegger wants to explain how something like subjectivity, in an ontological manner, is made possible. This he does in Being and Time and the lecture courses mentioned above (1925-27).This is a difficult issue, and it has taken a lot of twists and turns inthe literature. Dreyfus has discussed this with particular detail in his book, as have a number of other commentators (Crowell, Boedeker Dostal Jr). There doesn't seem to be a concensus on the subject. I think the short answer is simply: Dasein is the way in which entities which care for being relate to being. This is broad enough to include both for self-interpretation without assuming self-interpretation to be at the basis of ontological inquiry. The 'I' is not disposed of, but neither assumed as the ontological clue.
The assertion that it is I who in each case Dasein is, is ontically obvious; but this must not mislead us into supposing that the route for an ontological Interpretation of what is 'given' in this way has thus been unmistakenly prescribes. Indeed it remains questionable whether even the mere ontical content of the above assetion does proper justice to the stock of phenomena to everyday Dasein. It could be that the "who" of everyday Dasein just is not the "I myself" [B&T, Pg. 150, 115]
As long as we don't take the fact that Dasein is an entity to imply that it is a mind with private contents first and foremost, I think all is in good order.On the general note, I think this topic can ammount for some very good questions. A good question that follows is whether animals have a world ornot (Heidegger says NO), and whether his interpretation is fair. The text where this is discussed is The Fundamental Concept of Metaphysics(1929/30). The notion of worldhood is discussed all throughout his work,including the abovementioned lecture course and Being and Time.