I thought about that being what he was alluding to, but that's not just a Gingerbread phenomenon so it had me think he was talking about native apps. Even on Android 4.0, the stock browser remains static unless there's an OS upgrade. It would just be kind of cherry picking to just pick on Android 2.3 in that case when every Android OS version's stock browser (outside of those that come with Chrome), is dependent on the OS version. Just a bad idea that left it dependent on the OS after seeing how that went with IE. If he's referring to Chrome only working on ICS+, that's true, but not every ICS+ device comes with it (very few do), so most people won't bother getting it any more than they would look for an alternative browser on Android 2.3.
I haven't used iOS enough to say for sure, but isn't Mobile Safari also tied to the OS version in a similar way and can't get updates without the OS being updated?
It isn't just a Gingerbread phenomenon, but Gingerbread is one of the most obvious examples of the phenomenon. It looks like half of all "active" Android devices are running Gingerbread or earlier [1], and many of those devices will never have an OS update released. It happens with iOS too, but devices generally get updates for two years before they're stranded.
But you are correct--it is far from the only example. There are major problems with iOS's model, too--once your device has received its last update, you're stuck with its rendering engine forever! Browsers using alternate rendering engines aren't allowed in the store. With Gingerbread, at least people can install a browser with a newer rendering engine.
But in general, I think the "Gingerbread is the new IE6" claims are pretty hyperbolic. IE6 remained such a scourge because so many companies wouldn't upgrade it to maintain consistency and compatibility with their custom, non-standards-compliant internal web applications. In contrast, mobile devices currently aren't kept very long, and there's quick iteration. After a few years, those Gingerbread users still using their devices can get a new browser if they're insistent on using the device, and idevice users will have to get a new idevice. But those users won't be tied to using those devices by corporate policy. At worst, I think developers will have to worry about the Gingerbread browser and older versions of mobile Safari for a few years, not the better part of a decade like with IE6.
I haven't used iOS enough to say for sure, but isn't Mobile Safari also tied to the OS version in a similar way and can't get updates without the OS being updated?