Baby is synonymous with infant. Often, when people say baby (other than as a term of endearment), they are typically referencing an infant. I find that infant is a bit more descriptive than baby. Can we make "baby" a tag synonym of infant?
My initial thought was to agree with Karl. However, upon looking at baby, I see it has only been used 21 times. Every one of those would seem to do better with one of our age-specific tags (newborn, infant, or toddler).
I believe making baby a synonym of one of the age specific tags. I'm not certain which tag makes the most sense, though. I'm inclined towards agreeing that infant makes most sense, so I consider a vote for this answer to be a vote for making baby a synonym of infant.
However, since there are other synonym possibilities, I'll post separate answers. If you feel one of the other choices is a better synonym, don't vote for this answer; vote instead for the other.
Regardless of which synonym we choose, this would require some cleanup, first.
In some cases, baby replaces one of our age-specific tags, in which case the question isn't tagged appropriately (e.g. the question is about an 18 month old, and it is tagged baby, so someone searching the tag toddler won't be able to find it).
In other cases, it is present in conjunction with an age-appropriate tag.
In a few cases, it relates to a baby that has not yet arrived: Querying a database for baby names within popularity range? and Adopt before giving birth or give birth before adopting?.
In any case where baby is used with an existing age-specific tag, it should simply be removed.
In any case where it is used instead of an age-specific tag, it should be replaced with the appropriate age-specific tag.
In the two examples for a child who has not yet arrived (the database on baby names, and the adoption vs. birth questions), I think the tag should be removed altogether.