It could be teething or self-soothing. And no, "self-soothing" is NOT just at a certain age. Self-soothing by sucking fingers/thumb even occurs in the womb.... a 'baby" in utero does this too. It's a "natural" way to soothe for a baby.
My son, at the age of 6 months, took to a pacifier. Before that he didn't. He had "discovered" one of my daughter's pacifiers laying around in our clutter (LOL) and started sucking away. It doesn't bother me. Also at this same age, my son took to a "lovey"... he "adopted" his stuffed cow and it is his constant "pal."
Research "self-soothing" online... there are many opinions on it. Many child experts don't feel it is "alarming." And some will. It is also cultural and attitudinal. In our culture, self-soothing and getting "attached" to a lovey or the breast is frowned upon.
The thing is, she will either grow out of it, or the Parent will intervene and stop or "wean" the child off of their "soothing" habit. Or it will bother the Parent and then the Parent will do something about it. Or a Pediatrician will say something about it.
It all depends on what your take on it is. There are SO many different opinions on this, both professionally and as a Parent. Just go according to what you think will be best for your child.
I was a thumb-sucker as well. In those days as least, it was not such a big deal if a baby/child self-soothed or got attached to a "lovey." I stopped on my own, as most children do. It really depends on comfort level...and on the parent's comfort level or if they feel uncomfortable with outsiders making comments. Which is what happens sometimes.
Sorry I don't have a magic answer... but just some thoughts. Just go according to what you think is best.
All the best,
Susan