This one's short but important - there's been a change in SharePoint 2013 when you install a custom claims provider. The IsUsedByDefault property is now set to False by default; in SharePoint 2010 it was True, which means as soon as you installed your custom claims provider everything "just worked". You will now have to go and change this, either after the fact or as part of your claims provider receiver assembly, in order to have your claims provider start working.
Also, I'm finding that those same tools used in SharePoint 2010 - the SharePoint Claims project with the HttpModule and web part that lists your claims, the ClaimsProviderActivation tool, that enables and disables a provider - are still quite handy in SharePoint 2013. As a quick plug to help you get started I'm attaching a ZIP file to this post with rebuilt versions of those tools that work with SharePoint 2013.