Mobile Safari Browser and HTML5
Mobile Safari is HTML5 capable since v1.0, so since 2007. The exact implementation details for various specs changed over time obviously.

At the moment there is no browser on the planet supporting everything mentioned in the HTML5 spec. That's because it's a working draft, not a finalized specification.

There're a lot of components forming what we call HTML5 and the site you're mentioning needs a specific feature which Mobile Safari doesn't support. So the error message is a bit misleading and user unfriendly. You can see an overview about support for all HTML5 features here.