I fixed the problem. To use the ASP.NET Session State in the dedicated or co-located Azure caching, the following session provider needs to be setup in the web.config:
<sessionState mode="Custom" customProvider="AFCacheSessionStateProvider">
<providers>
<add name="AFCacheSessionStateProvider"
type="Microsoft.Web.DistributedCache.DistributedCacheSessionStateStoreProvider, Microsoft.Web.DistributedCache"
cacheName="default" dataCacheClientName="default"
applicationName="AFCacheSessionState"
/>
</providers>
</sessionState>