I think it's pretty similar for kids who aren't twins but are close enough to share interests, in that after the first few days, the toys generally get shared and nobody knows who they were for originally. I find that tends to be the case, with the rare exception of a special toy that one kid really gloms onto. But most of the toys, unless they are really age-specific (only my oldest uses the woodworking kit, only the toddler plays with the Elmo stuff), just get played with by everybody.
I love the King Solomon comparison. 
This is not specific to twins but, do the kids really not know who they originally were for?
My sisters and I all have birthdays within the same week (in different years) and we pooled our toys (they almost all lived in my baby sister's room, for a variety of reasons) but we were WELL aware of what belongs to whom.
In fact, my dad got out the Fisher Price record player for my daughter on Thanksgiving and mentioned he didn't know which of us it belonged to--3 of us said in unison "Dawbs". And we all knew that the hobby horse was baby sis's.
(We didn't tend to mind group gifts--although the older we got, the more it became an issue--we had some shared-purchase clothing that we debated fiercely about when it came time for some of us to do vacations/college/etc)
Well, it probably depends on the item, really. With the dolls? For the most part, after a while, nobody really knows which doll belonged to who anymore. Same with the stuffed animals. The exceptions are the ones that were really special to a kid, or that were bought within the last year or so. There are definitely a few of these (especially stuffed animals for my oldest, as she's a big stuffed animal fan, whereas the second is a big doll fan).
A lot of toys are sort of like... sets. Like my girls have these Stella Squeak toys. They have little plastic houses and furniture, and the characters are these little toy anthropomorphic mice. One kid got one house, one kid got another. Then there were various smaller sets. The kids *sort* of remember who got each house, but nobody remembers for any of the smaller sets or the mouse characters. LEGOs, of course, all end up in the LEGO bin once they've been put together a time or two.
And there are many toys that, while they *do* remember who it belonged to, they're no longer possessive of it after a while. My oldest *longed* for this Rapunzel doll back when she was 3. She finally got it for... maybe Christmas when she was about to turn 5? So she had wanted it for something like 2 years. Now she's about to turn 7, and she doesn't mind if a sister plays with it, although if you ask the girls, they know who it belongs to. Jenny will refer to it as "Emily's Rapunzel doll," but she'll still play with it without Emily objecting.
We have a set of tiny little princess dolls with removable clothing. These were Emily's favorite toys in the whole world, once, and a bunch of them were Christmas/birthday presents. I doubt she even remembers. At most, she might remember the one that she bought last, because she bought it with her first savings from her allowance.
Charlotte tends to claim any toy she wants to play with as hers, "From Grammy and Grandpap!" She doesn't know what she's talking about.
Half the stuff she tries to claim as hers was Emily's when Emily was a toddler. Emily, of course, has grown out of Elmo, so she doesn't mind Charlotte taking the stuff over... but sometimes it's a point of contention between Jenny and Charlotte and neither, obviously, has a strong claim to it.
I do find, though, that when stuff is new, it's helpful to know whose is whose. For instance, one year, my mom bought all three of the oldest a Pascal toy (a chameleon from Tangled). I ended up marking them (a dot with a Sharpie under the tongue) so I could tell which each belonged to. Partly because they'd find one, not know whose it was, and fight over it. Partly because if we knew *whose* was missing, we had a better chance of figuring out where it went.
I think, though, that the memories fade with time and interest. My kids are still little enough that after a few years, they aren't interested in the same kinds of toys. So they don't really remember that it was theirs or even care. I could see caring more if they were older, and the toys were older and thus being interesting and desired for longer.
