I think so. Most other ad networks very specifically deny you the ability to publish across your network of sites.
Google does have something that they're not telling us about which allows them to identify sites and owners. I run four sites all on different IPs (same block) but they all show up as being related to each other when I do a "related sites" search.
You can use same AdSense code on different sites, and yes they can be of different topics. Google is pretty good at visiting those sites ASAP and serve target ads, however make sure you do not put ads on any gambling or other sites that Google does not want in AdSense network.