Babies that age ARE attached to the mothers. Those who must be placed in daycare have little choice but to adjust; but it's very "standard" for a baby in the age range of a year old ("plus or minus a couple of/few months for some") to be extremely attached (wrong word, because all normal children are "attached" to their mothers, but you now what I mean) to his mother. In the area of close to a year old (or so) they become aware that they are separate people from their mother. Still, they're so young they don't really realize that if their mother leaves the room she isn't "gone forever". That stage can start a few months before a year old, but children can vary a little with regard to exactly when one stage starts and how long they're in one.
(With the one-day when she was four months old, there's a good chance she was just being cared for by someone who wasn't good at keeping her as secure as a baby needs to feel; and there's a chance something was going on where she was that was just frazzling to her (or she couldn't sleep for some reason or was otherwise uncomfortable). In other words, that one day doesn't particularly mean much in terms of what's going on now.
Besides being not far from a year old, there's the chance that she's a little more likely to seek the safety of your lap when other children are around. Toddlers that age aren't all that sure on their feet, and they're smaller than all the other children (who are often not all that careful around them), so babies that age can feel most comfortable (at least sometimes, depending on how much activity is around them) being with their mother.
Two is generally the age when children become very aware of wanting to be independent and do their own thing. Right now, she's still at an age when her Mommy is still very much her world. It's a great time to keep "growing" that bond, because when she's closer to two and past her second birthday she'll want to do her own thing more. I found fifteen months old to be among the most challenging time with my toddlers, not because they weren't "easy babies", but because they had a more developed stamina and energy level than earlier, but remained "clueless" in ways an eighteen-month-old or two-year-old isn't.
As for the blankie thing, not all children get all that attached to a transitional object; and even though many do, the age when it's "a big thing" for them can be different from child to child.