Testing Kits Are Available
There are asbestos testing kits available for home use. Unless something has changed in the last 20 years, asbestos testing relied on microscopic examination of the material, looking for the asbestos fibers. As of 20 years ago or so, there was not chemical testing for the presence of asbestos (unlike the testing for lead in paint). I believe the test kits all require a sample of the suspected material be collected and placed in a pouch for mailing to the testing lab. You can Google Asbestos Test Kits to see what might be available.
The prudent thing to do is assume any suspected material is asbestos and handle accordingly. In commercial work dealing with known asbestos, this would mean using respirator designed for asbestos, rubber gloves, and a paper suit. After use, all of these items should be put into a plastic bag along with the removed material, the bag sealed and placed inside another bag, and it sealed. Labels would be attached identifying the material as being asbestos. This would have to be taken to a hazardous material disposal site, not thrown into the trash can along with normal trash.
Asbestos is most hazardous when it is "friable", that is, the fibers break off and become airborne. Wetting the material down and keeping it wet reduces exposure.