No, you cannot do this. You just need to install it in a separate statement:
var obj = new MyObject(); obj.Test = obj.Id;
The right side of the property in the initializer of the object is just a normal expression that does not have a built-in connection with the initialized object.
If this is what you usually want to do with one particular type, you can add a method:
public MyObject CopyIdToTest() { Test = Id; return this; }
and then use:
MyObject obj = new MyObject().CopyIdToTest();
or with other properties:
MyObject obj = new MyObject {
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