There have never been any such limitations in the SV standard. You can even pass class handles through ports of a module.
The key limitation with classes is where you are allowed reference a class member or method - only from procedural code.
So that means you can use a continuous assignment to assign one class variable to another class variable, but you cant continuously assign a class member to another member.
module top #(type VAR_TYPE);
VAR_TYPE a,b;
assign x = y; // legal
assign a.x = b.x; // illegal
endmodule
Note: you do not need the parameter keyword in the parameter list, and you should not assign the parameter a default if you want to require the user to provide an override. You will get much more meaningful error messages if the user fails to provide an override than leaving the type as integer.