She is only 6.
I have 2 kids, that when younger, were shy.
Anyway, I think your daughter will "blossom" more and be more "social" or whatever you want to call it, each year as she gets older.
But then again, some kids are just this way.
AND no kid, has to be friends with all the kids nor talk to all the kids all the time or everyday.
I work at a school. Some kids are just more introverted. Some are extroverted. And quite frankly, the "shyer" ones do just fine. They are not meek and weak and unable to navigate themselves. They just are more introverted, and they choose... their friends. Versus some kids who are so insecure unless they are among a huge group of other kids and having to be among a group of kids because they think that makes them "popular."
So there are pro's and con's for each, personality type or social tendency.
Just keep nurturing your child.
Teach her to KNOW herself and who SHE, is.
As I said, my kids were shy when younger. Especially my daughter. But so what. All along, I nurtured THEM as individuals. Taught them to KNOW themselves and who THEY are. And by the time they were in Kindergarten, they had a solid sense of who they are. Versus other kids. And when a kid knows, themselves... they also are skilled in navigating amongst social situations and in choosing friends. They know, how to discern... things. And this is a plus.
So, EVEN if a kid is "shy"... it is not detrimental... nor a "bad" thing. Because, many of these kids, are keen observers and keen about character types in others etc. and in who they talk to or interact with. At least my kids were and are. EVEN if they were shy, when younger.
Now at their ages, they are not so shy. But they are very good at gauging all sorts of personalities and social situations. Of which, I nurtured that in them.
I work at an Elementary school.
At this age, Kindergarten and even 1st grade, MANY kids... are SHY when speaking or in front of others. It is common. YES they "whisper" and talk softly. That is why, there are things like show and tell or reading aloud.... so that kids are introduced... to speaking etc. It is NOT an inherent skill. It takes, practice. Over, time.
When/if kids ask how come another kid doesn't talk, I say, not every kid has to be chatty. He/she is shy. Its no big deal. It is just, fact.
I don't play it down or play it up.
Its life. It is a fact. Some people are just not as talkative. And it is okay.
Just because others are more talkative, it does not make them better or worse than another. It is just, everyone is different.
I even know some 5th graders, that are more shy. BUT it is their personality. It is their, choice. They have a few friends. But they are happy. They CHOOSE their friends. They know who they are. They are not at a loss...socially or per how other kids like them or not. They are normal good kids. They are themselves. They don't get caught up into the drama-filled-popularity trips and cliques, like the other kids do. And they are actually very high functioning stellar kids, who are mature and are very self-reliant and disciplined and they navigate themselves, amongst "icky" kids, very very well. They are also very sage like, in their observations of others and situations.
And some of them, are actually the "leaders" in class. Even if they are not the extroverted gad about type, kids.
And look, just TEACH your daughter... HOW to ask for help, how to speak up, etc. EVEN SHY KIDS, can do so. But you need to nurture that and teach the child.
You can always ask your Pediatrician, what he/she thinks.
With my kids, I never made a big deal over if they were shy or not, nor compared them to other kids.
And they had no hang ups about it.