“Auschwitz” was a camp complex based around the town of Oswiecim in Poland. The name itself was a German rendering of the town’s name.

This complex contained, at its largest, forty subcamps. This was massive.

Referring to any of those subcamps individually could be done either by number or by name. The death camp section, for instance, was specifically Auschwitz II or “Birkenau” - the “birch meadow” camp, so named because of a clump of birch trees at one end.

If you were referring to “Auschwitz III Buna-Monowitz” or “Raisko”...both of these are technically still Auschwitz. They’re smaller subcamps in the Auschwitz network, but they’re still Auschwitz.

The number specifies which specific subdivision within that group you mean - the central camp I, the death camp and crematoria complex II, or the scattered subcamps all over the surrounding region as III.