Us too, Mashi. DD has bun whenever she wants. For us, it is a comfort item, not just a sleep item. Lo's are so attactched to their lovies, and IMO that is a GOOD thing. They can soothe and calm even when Mommy can't. The only rule there has ever been is that bun stays home..from day one...we kiss it goodbye and go. She never has asked to bring it, since it has just never been an option.
It is hard to ask our lo to rely on something for comfort, but then try to limit that item....kwim? I think, they need it when they need it, it's a healthy thing, so we just let dd self-regulate! If she asks for bun, it is because she is tired, upset, or just feeling snuggly. I love that she has bun...she is a member of the family! While we began using it for sleep (we would tuck bun between us at wind-down), she soon came to see it as not only for sleep, but for any time that was quiet and relaxing..(books, tv, etc).
IMHO...(I am a huge lovey-advocate, obviously), I don't see how denying them the thing they have grown to love could "not" be confusing and frustrating for them? Again, jmho.