If your case is an older Brady Case the internal covering will be Baize not felt. Now the traditional glue used to fix trade labels is pearl or rabbit skin glue for a couple of good reasons firstly they do not travel through the paper when cold/dry causing staining over time, instant fix because they are used hot and secondly they can be reversed with steam or hot water and the glues age is no limit to when this can be done. Though a more complicated removal if it is a leather trade label and finally they have stood the test of time that is why they are still available and used today.
What professionals use in many cases I have found is chosen for what good for them be it financial or expediency!