What you are saying, my car can take me to Walmart, but I can't get to Safeway.
In order to get to a site (you are getting video from an Internet, outside of your home place) the TV needs an APPLET (a piece of software that talk to that particular site). You need a separate applet for EACH site. You probably haven't noticed this because your remote has a NETFLIX button and everything is automatic. Your TV may not have an Amazon applet.
HERE is a list of compatible devices that can get Amazon videos.