Jewish people emigrated into countries and stayed in their own small groups with languages, such as Yiddish, that were not used by the rest of the country they inhabited (except in borrowed language). No, Judaism is not a nationality, and in the subject of language it is not treated as such.
When surnames were becoming hereditary, Jews did not inhabit one single country but still used a common language, Hebrew (mainly as a second language of course). This itself would have led to names such as Cohen which were directly taken from Hebrew. These names, as well as names derived from Yiddish (a mix between Hebrew, German and Slavic) could be described as nothing else but Jewish, they aren't Israeli obviously (there are a few Israeli Hebrew names taken up in the last 100 years or so), so what else would you call them.
Then you have names that are taken directly from the language from which the people lived (eg. German). Names such as these are often used by both German gentiles and German Jews because they have dual etymologies, a German one and a Yiddish one that were similar because of the obvious similarities in language and therefore ended up as the same surname.
The truth is, if your surname is actually considered Jewish in the linguistic field,then it is because that it had Yiddish (which is often similar if not identical to German words) or Hebrew. Names that have are not descended from Yiddish or Hebrew words and are still considered Jewish are probably called that because Jews are the only ones that took it as a surname.
Jewish names (of which there are many dispite your views to the contrary) are taken from German words, Polish words, Yiddish words and Hebrew words. Jewish names are fairly unique in the sense that they cannot be put into the simple categories of nationalities. I'm sure that names that are mainly used by Christians in non-Christian countries are referred to as Christian names.
I don't really understand why you should take such exception to this categorization of surnames, it isn't done to segregate people but simply because often the names that are used by Jews (whatever their language of origin) are only used by Jews. You are not being 'lumped in', it is a necessary category of names borne out of unique circumstances.
I hope this helps you understand just a little bit (but I am no expert) as to why this categorization is necessary. :)