Welcome to the community, @arielaronica!
Unfortunately, Airtable provides no way to automatically get a non-formula field (such as a single select field) to update itself, unless you have already grouped all of your records by that field, and then you manually add a new record into the particular group where you want the child’s record to go. In that case, Airtable will helpfully automatically type in the right value for you.
In other words, if you already know in advance what group a new child belongs in — and your records are already grouped — you can manually add a new child into the correct group, and Airtable will add the group value for you.
But that can be a lot of hassle and probably not worth the headache, because you would need to know the child’s group BEFORE you even type in the birthday. You would need to know it before you even type in the child’s name. And then you would have to scroll through your entire list of records (or collapse & expand the groups), just to get to the right group where you would want to add the new child.
So that’s probably not worth the hassle, but maybe it is if you already know the groups ahead of time.
Getting into automation:
When a record’s birthday field changes, Integromat will automatically update the Age Group for each record. This would be my preferred way of doing this… it’s easy, it’s quick, it’s free (as long as you stay within their limits for number of records), it’s fully automated, and it doesn’t require any coding knowledge. Integromat is pretty amazing. (BTW, if you have a budget for this project and you need to hire someone to help you setup Integromat, feel free to send me a private message. I am a professional Airtable developer.)