I found a site that is using my name "xyz" as the name of their site. the name of my site is xyz.com the name of their site is xy.com
I launched my site before they had any site, however they reserved their domain name before I reserved mine. The name of their domain name only partially reflects what they are calling themselves - as the example above shows. We both are offering completely different services even though are name is the same. I don't have my name trademarked and I'm pretty sure they don't have theirs trademarked either.
Now we both have confusion amongst visitors as to who is who.
You say you operate in different markets, that means you are both entitled to use the name (apart for some special circumstances).
Lets say you sell food and the other company provides healthcare services, there should be no problem with you both being called XYZ. The thing about trademarks is that they distingush a particular type of product or service, you can't own a trademark across all types of business unless your business is remarkably diverse.
Examples that show this are companies that use initials - should there be only one company allowed to use a set of initials? of course not, but if you try to sell food under the initials KFC you will be in trouble. Kirremuir Fencing Contractors would be able to use KFC in their line of business though.
You might want to put a discreet link on your home page, along the lines of "Are you looking for xy.com?", with a link to them. And in case your users are having the same problem (hitting the other site instead of yours), you might send a polite request to xy.com that they do the same.
I've had occasion to land accidentally on an unrelated site because I'd mis-guessed the URL of my intended target. I always appreciate when the mistaken URL provides a link to the intended one. It probably saves them from having to deal with a lot of unnecessary e-mail queries, too.