I need to know whether a custom object is only a custom object(in classic menu > create > objects), or if it has custom metadata types and custom settings as well. This is because I need to use the license platform starter.
link of price and restrictions for licenses: https://www.salesforce.com/editions-pricing/platform/
I will be using custom objects, custom metadata types, and custom settings in my designs.
You should be fine, poke your Salesforce Account Manager just in case.
Custom Settings and Custom Metadata aren't "real". Normal users can't access them, only admins can. But even admins can't say run reports on them, they don't show up in Storage Usage, you can't set Profile-level access, field level security (and Profile is where the License really starts to play).
Easiest would be to just try it out, create 1 such user and see if they can create your custom object (and have some flows, validation rules etc run on create with logic based on custom setting / meta)?