It certainly does. There are folks who study the manuscripts of any given piece of literature that has been copied, like the Bible, since antiquity. Most of them just go by Classicists. Take any intermediate-advanced Latin class and you'll find that the texts we use now are the result of years of manuscript study. Even then you will frequently find discrepancies between manuscripts that we're still not sure of the correct way. This goes for any author from Vergil to Marcus Aurelius.

Keep in mind that the concept of "field" is entirely constructed and not completely reflective of the work being done. There are simply so many more copies of the Bible, and much longer ones at that, than other texts, so it's more likely that someone will specialize just in studying it.

Most of this work is done by classicists these days, as u/CommodoreCCo says. It used to be its own (sub)field called philology, which focused mostly on producing useable, carefully edited printed issues of ancient / medieval texts. These works do things like the biblical criticism you describe: comparing different surviving manuscripts, trying to reconstruct the original texts from which they were copied and resolve any differences if the texts disagree with one another, trying to reconstruct the histories of various manuscripts and how / when they diverged from one another, and whether there are any interpretive / translation issues that might influence our (mis)interpretation of their contents.

These days, philology has fallen somewhat out of fashion, and most departments require scholars to produce and publish original research / arguments about texts rather than critical editions of texts; it's difficult to get tenure in a history or classics department if the only thing you've produced is critical editions. This is something of a shame, because making a good critical edition requires just as much work as writing an original argument, and folks like me who rely on texts for our work, but don't have time to produce our own critical studies of everything we use, rely heavily on the legwork generations of philologists have done for us. But this work is undervalued.

As a consequence, many historians (especially us medievalists) have to rely on older editions made in the 19th century. These editions are generally very good (19th century scholars really knew their latin, etc), but it would be beneficial of more people were able to revise some of the older, or more problematic, ones. There are also a fair number of less popular texts for which editions have never been made, but probably should be.

A few groups like the MGH are producing new or updated critical editions, and some tenured scholars are doing good philological work. But it's a field worth more investment by future generations.