An AG corporation has stocks in order to track who owns how much and also attach different economic and voting rights to different classes of stock. The other way to incorporate a limited-liability company in Germany is the GmbH, which tracks ownership directly in the articles of incorporation, but are in other ways subject to way lower management, disclosure and accounting requirements. So the AG is mostly useful if you want it to be easy to change your ownership structure, if you for instance raise capital from new investors, issue employee shares, change cross-ownership within a conglomerate or go public some day.
Why Carl Zeiss is an AG I don't know. The West German Carl Zeiss was re-formed as a GmbH in 1946, but had changed to an AG by 1973. The East German Carl Zeiss was turned into a GmbH during reunification and then split in two. One part merged into the West German Carl Zeiss AG and the other is now called Jenoptik. Jenoptik was converted into an AG in 1996 and went public in 1998. AFAICT Carl Zeiss has been privately owned by the Carl Zeiss-Stiftung since 1889, except of course for the temporary East German part.
Why Carl Zeiss is an AG I don't know. The West German Carl Zeiss was re-formed as a GmbH in 1946, but had changed to an AG by 1973. The East German Carl Zeiss was turned into a GmbH during reunification and then split in two. One part merged into the West German Carl Zeiss AG and the other is now called Jenoptik. Jenoptik was converted into an AG in 1996 and went public in 1998. AFAICT Carl Zeiss has been privately owned by the Carl Zeiss-Stiftung since 1889, except of course for the temporary East German part.