That is a brace. Not only does it support the seam, it adds significant strength and stiffness to the back. Left loose, it can be the source of a really annoying rattle. It should be re-glued.
Assuming that the guitar in question is of decent quality the strip is correctly cut to length. Do not trim.
The most difficult part of the repair is removing the dried glue. It is (probably) on both the bottom of the brace and the inside of the back. Small scrapers, like those made from razor blades, shim material, or even old palette knives are the best choices. It might appear that using sandpaper would be a good idea. Do not do it! Sandpaper can tear and get lodged between the brace and the back. That will create three times as much work because it will have to be removed before gluing. A damp rag can be used to clean up any residue when scraping is done.
Once the glue is removed and cleaned up devising the clamping rig is the next problem. This is a perfect job for a
go-bar table. Most luthiers and repairman will default to any number of clamping arrangements. One of the better ones will be to use a
brace repair jack(s) between the top and the brace. When you have chosen the clamping method, a dry run or two are in order.
Either white or yellow glue are good choices for this job. White glue has an open time of roughly seven minutes. Yellow is about five. Apply the glue with a spatula to the brace and back. Place the clamps and apply pressure on the brace until some glue squeezes out. Clean the glue with a damp rag. Continue the clean up process for a couple minutes. The glue will stop oozing. Allow to cure overnight. Speaking of glue, if the glue on hand is older than about six months, throw it away and buy a new bottle. This is not a job you want to do twice.
Remove the clamps, do some final cleanup. String it up and play.
This job will take an hour for a skilled tech, maybe longer depending on whether or not the initial cleaning process shows more problems or creates more as the brace is lifted. This task requires patience. It also requires some woodworking skill. It will probably require some tool purchases. All of which point to the cost effectiveness of using a professional to perform the repair.
In any event, if think changing a tire is working on a car, have to look in kitchen drawers to find some tools, or think a tool is a piece of software, find a good tech to do this one. You'll be glad you did.