You also cant Trademark a domain name. You can only trademark the contents of that domain, ie, the website itself, not the domain name.
A domain name is rented, not bought. You must renew your domain name at certain intervals, or it goes up for sale. You can't "buy" a domain name. You only own it for the time of your rental.
Now if you "cloned" their site, like i see so many requests for here, you may have a problem. Cloning means that you "copy" the look and feel of their site exactly, and sometimes not exactly. I have seen cloned sites that change only a few things, but overall that is not only tacky, but legally wrong. You may get legal action against you for that. But if you did not "clone" the other site, you have no worries. I have looked at both sites and you didnt clone it.
They see you a rival competition and therfore feel threatened that you may take away business, but they do not have a legal leg to stand on. They are trying bully tactics to get you to take your site down. Do not fall for it. You have every right to run your site. Competition is what drives the market place.
Stand your ground and they will go away
IMHO (researched of course)