I have that same lab coat. It is made by Historical Emporium, they make a lot of historical replica clothing. Everything I own from them is extremely well made.
I believe it is, at least for Firefly. Low budget and existing sci-fi armor the team had access to.
For anyone who doesn't know, a deep dive into re-used props and costumes in movies can be a pretty cool and uselessly informative way to kill an afternoon
Apparently the reason Firefly and quite a lot of other productions used it was that it was the last large set of decent quality sci-fi costume armor produced that wasn't TOO iconic like with storm and clone troopers.
Basically immediately after Starship Troopers if you needed more a few sets you'd just do it digitally, so all the down stream productions just traded that set between them and modified it bit by bit.
Supposedly some bits and pieces are still in use but they've been cut down into components.
More than likely this is the case. A lot of costume designers on a production will rent stuff from a costume house, or they may own their own costumes and rent those to the production.
When I did locations for the industry I rented my heaters, trash cans, trash trailer, and generators to productions so they did not have to buy their own.
Like Joss Whedon reusing the science uniform from Alien Resurrection for Dr. Horrible
I have that same lab coat. It is made by Historical Emporium, they make a lot of historical replica clothing. Everything I own from them is extremely well made.