Multiple partial class definitions that refer to the same class must all exist within a single assembly. In your example above during compilation the meta-data should be baked into the class and after compilation the class is whole, comprising of all the parts. Partial classes are a means to split the definition of the same class into multiple files.
See here for an exhaustive explanation but note the following:
All partial-type definitions meant to be parts of the same type must be defined in the same assembly and the same module (.exe or .dll file). Partial definitions cannot span multiple modules.
This link here explains that
In general, ASP.NET creates an assembly for each application directory (such as App_Code) and one for the main directory.
For your case, although the partial classes are in the same project and the same namespace they are not being compiled into the same assembly.