I bought an anti static mat that came with a banana plug connector and an alligator clip that can be mounted on the banana plug. Can I connect the banana plug connector directly into a US ground socket hole? Or should I buy a ground plug adapter like this instead? Link to ground plug adapter on Digikey
Just use a UL listed ground plug adapter
Don't mess with AC mains. Anything that connects to North American style mains, needs to be built to UL White Book standards and UL-listed as equipment. (Or equivalent from CSA, ETL etc.)
"RU" is a Recognition for components, and is not equivalent.
Random jumper plugs are not built to go into AC mains ground pins. They either are not UL Listed at all, or their instructions do not authorize that use.
The police aren't going to arrest you if you do this, but your insurance company will laugh at your claim.