I think this question would be more suited for Biology beta, but since it's here, I'll try to keep the answer as lay as possible.
Albumin, like you read on Wikipedia, is a large group of proteins, which are present in all kinds of organisms, including your own blood. (Actually, albumin in your blood has a very important function - it binds small molecules, such as ions or medicines, making them harmless to the body.) Albumin from one organism is not just one kind of polypeptide, each species produces many kinds of albumins. Therefore, albumins are present in your blood, in egg-white and in milk, but they are somewhat different proteins.
About lactose and albumin intolerance. There are two kinds of resposne - intolerance and allergy. Intolerance baisically means, that your body lacks mechanisms to break down a certain particle (e.g. humans are all celulose-intolerant), so eating intolerable food usually doesn't hurt more than some belly pains or a day of diarrhea. Generally, all proteins can be digested, but not all carbohydrates, so it's common to be lactose-intolerant but have an allergy towards egg-white.
Allergy on the other hand means, that your body is trying to "kill" the intruder, even if it's as harmless as a molecule of foreign albumin. Every molecule that triggers this immunologic response has haptens. These are in fact small parts of the molecule, that are "visible" from the outside. Since all organisms produce different kinds of albumins, the haptens of those albumins are different. That's why albumins from one organism may trigger allergic response and from other are tolerated.
Now, given all this basic information:
yes, people can be sensitive to one implementation of albumin but not sensitive to another;
yes, people may be lactose tolerant but intolerant to milk albumin, though "tolerance" is probably not the right word here;
and yes, there might be people who think they are lactose intolerant, but are actually milk-albumin intolerant, but it's not very likely.