Miriam Bulwar David-Hay
Yes, Auschwitz was the only camp -- or to be more accurate, camp complex, because it was huge -- that tattooed prisoners with numbers. There was another, much smaller camp (Mielec) that tattooed prisoners with the letters KL, but that's clearly not what we're discussing here. Auschwitz prisoners were often sent on to other camps, where they would receive new numbers, in the form of patches on their uniforms, but obviously the tattooed Auschwitz number would still remain, and this may have led to confusion later about whether the number was from Auschwitz or another camp. But it was Auschwitz only.
There are several reasons why you may not have been able to find their names in the Arolsen database. Firstly, try some spelling variations on the surnames, first names only, place names only, etc., as the Arolsen system is not as good as some others in "sounds-like" searches. Secondly, know that not everything Arolsen has is online, so if you can't find what you want in the website, fill out their online inquiry form and see if they can find something in their offline records. Thirdly, know that the Germans destroyed a large portion of the Auschwitz documents when they evacuated the camp, so it is possible those records may not have survived. Fourthly, try searching the Auschwitz museum website as they have some prisoner records online, and, if you can't find anything, try writing to them too as they also have material that is not online: http://auschwitz.org/en/
Finally, try searching both the U.S. Holocaust Memorial Museum and the Yad Vashem Holocaust Museum and Memorial websites, as each has extensive records online, some of which overlap, but some of which only one or the other has.
Best of luck,
Miriam Bulwar David-Hay,
Professional writer, editor, proofreader.
Professional translator (Hebrew & Yiddish to English).
Certified guide, Yad Vashem Holocaust Museum and Memorial.
Long-time family history researcher.