Bergkamp and Ozil both have outrageously good technique, Ozil probably has the edge on Bergkamp in terms of physical gifts (pace and agility), both have good tactical nous. Bergkamp was more of a leader than Ozil, who can be a bit tempermental, and was more of a big game player in my view. Bergkamp was perfectly suited to his role on that Arsenal side though, and that position/role was slightly different than Ozil's.
Let Ozil stick around for 5+ years and win some trophies, and then let's measure his impact at Arsenal compared to Bergkamp's. Ozil is on track to have a better season than Bergkamp probably ever had, but longevity is important in football. Ozil's first two seasons were underwhleming on the whole for me, conceding that injuries played a large role in that. Kaka was on Messi's level for two years in Milan, and was barely up to Madrid's bench standard three years later.
Comparisons of Ozil with Zisou are off the mark atm, Zidane is the undisputed best number ten of his generation. Zidane is a Ballon d'Or winner, was the fulcrum of the Galacticos side in a way that Ozil was frankly not at his time at Real Madrid.
To be fair, Zidane was a tier above Bergkamp as well in all honesty.