I couldn't find any information on this doing a search. I know that custom settings count against the custom object limit, but wasn't sure about custom metadata types. This is important as we can only have 20 custom objects in our org.
I don't think they count against your custom object limits. Custom metadata types have their own limits for your organization.
Custom metadata types per organization: 100 This number includes all types developed in the organization and installed from managed and unmanaged packages.
NOTE: Custom metadata records in certified managed packages that you’ve installed don’t count toward your organization’s allotment. However, custom metadata records that you create do count toward it. This rule applies regardless of whether you create records in your own custom metadata type or in a type from a certified managed package.
Please refer this article for more information on limits. Custom metadata types limits and considerations