I am making a constructor in Python. When called with an existing object as its input, it should set the "new" object to that same object. Here is a 10 line demonstration:
class A:
    def __init__(self, value):
        if isinstance(value, A):
            self = value
        else:
            self.attribute = value
a = A(1)
b = A(a)#a and b should be references to the same object
print("b is a", b is a)#this should be true: the identities should be the same
print("b == a", b == a)#this should be true: the values should be the same
I want the object A(a) constructed from the existing object a to be a.  Why is it not?  To be clear, I want A(a) to reference the same object as a, NOT a copy.
self, like any other argument, is among the local variables of a function or method.  Assignment to the bare name of a local variable never affects anything outside of that function or method, it just locally rebinds that name.
As a comment rightly suggests, it's unclear why you wouldn't just do
b = a
Assuming you have a sound reason, what you need to override is not __init__, but rather __new__ (then take some precaution in __init__ to avoid double initialization).  It's not an obvious course so I'll wait for you to explain what exactly you're trying to accomplish.
Added: having clarified the need I agree with the OP that a factory function (ideally, I suggest, as a class method) is better -- and clearer than __new__, which would work (it is a class method after all) but in a less-sharply-clear way.
So, I would code as follows:
class A(object):
    @classmethod
    def make(cls, value):
        if isinstance(value, cls): return value
        return cls(value)
    def __init__(self, value):
        self.attribute = value
Now,
a = A.make(1)
b = A.make(a)
accomplishes the OP's desires, polymorphically over the type of argument passed to A.make.
The only way to make it work exactly as you have it is to implement __new__, the constructor, rather than __init__, the initialiser (the behaviour can get rather complex if both are implemented). It would also be wise to implement __eq__ for equality comparison, although this will fall back to identity comparison. For example:
>>> class A(object):
    def __new__(cls, value):
        if isinstance(value, cls):
            return value
        inst = super(A, cls).__new__(cls)
        inst.attribute = value
        return inst
    def __eq__(self, other):
        return self.attribute == other.attribute
>>> a = A(1)
>>> b = A(a)
>>> a is b
True
>>> a == b
True
>>> a == A(1)
True  # also equal to other instance with same attribute value
You should have a look at the data model documentation, which explains the various "magic methods" available and what they do. See e.g. __new__.
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