Having carefully looked at the logo under discussion I would venture the following opinion with this disclaimer:
1. I am not a lawyer.
2. This is my personal opinion based on a cursory reading of US Trademark and Copyright Law to satisfy my own informational needs for basic protection for my own names and logos.
A. The name in discussion has a similarity to eBid in that it incorporates the name *as an element* of the complete name for the site.
B. There are enough additional elements in the name that a "reasonable person" will not confuse ebid.net with ebid.nashville.gov
C. The artwork and color palette would be difficult to confuse with the current ebid logo and color scheme.
eBid has basic copyright protection of the name in the USA in the context of an online auction site thru 9 years of continuous usage. Additional legal right can be secured by registering the apropriate trademarks and copyrights. As has been stated the site owners or their legal representatives are the only ones empowered to secure these rights under US law.