Correct. There is no such thing as a "Jewish race". I was not born Jewish, but I am a Jew. I am not unusual, either.
Although most non-Jews may not realize it, people have been converting to Judaism for centuries. The classic example is Ruth the Moabite, a beloved Jewish matriarch and ancestor of King David, who was not born Jewish, but became a Jew by self-declaration and keeping God's laws:
"For wherever you go, I will go. Where you stay I will stay. Your people will be my people and your God will be my God." (Ruth 1:16)
There have been many periods in the history of the Jewish people where thousands of "gentiles" converted to Judaism and became Jews almost en masse, and they have been welcomed as a blessing. Frankly, I sometimes wonder how much of a blessing I myself might be to the Jewish people (dubious, at best :^), but I am proud to be Jewish, and that has everything to do with my people, not my race.
The notion of "Jewish ethnic purity" being some sort of goal of Judaism is a fallacy.
It is true that observant Jews want their sons and daughters to marry Jews, but race has nothing to do with it. Preserving the incredibly rich and ancient heritage, faith, teachings, philosophies, language and legacy of the Jewish people has everything to do with it.
This is not unique to Judaism, either. I know a few Catholics who don't want their daughters to marry non-Catholics. Does that make them racists?