The first possibility that came to mind was that this may be a "So-Called Dollar." These are roughly dollar-sized medals commemorating various events that were privately minted during the 1800s and 1900s.

There is a website devoted to these So-Called Dollars, mirroring the standard book on this topic by Hibler and Kappen. However, from a quick check of that site, I came across only one so-called dollar bearing President Theodore Roosevelt's name; that one was minted for the Louisiana Purchase Exhibition in 1904:

http://www.so-calleddollars.com/Events/Louisiana_Purchase_Exposition.html

Also, So-Called Dollars can't, by definition, be purely "presidential": they have to be associated with some particular event.

Therefore, another possibility - and a more likely one - is that this is simply a medal commemorating President Theodore Roosevelt, listing the start and end dates of his two terms in office.

There have been some official or privately issued medals, or other "exonumia" (non-coin items of interest to coin collectors), connected with nearly every (or all?) US President. These range from the formality of medals struck by official US Mints to the aluminum "Mr. President" game pieces issued by Shell Oil in the 1960s.

Is this 'coin' or medal still listed in that Bulgarian or Russian auction site? I didn't readily find it there.

If not, can you put up a photo or two - such as a snapshot with a digital camera or cell phone camera - at a URL where those of us who may wish to answer this question can see it? There are lots of free sites where you can post photos, if you don't already have a convenient place to put them; flickr.com comes immediately to mind.