How to make sympy acknowledge that 0/0 is a division by zero

As you noticed, the issue is that x/x evaluates to 1 immediately, so a subsequent substitution of x by 0 is not really happening.

A way to avoid evaluation is to pass evaluate=False flag to the product of x with 1/x:

>>> expr = Mul(x, 1/x, evaluate=False)
>>> expr
x/x
>>> expr.subs(x, 0)
nan    

The result is "Not A Number", meaning this does not evaluate to a number.

(In SymPy, all division is multiplication by reciprocal quantity: there are Add, Mul, Pow but no Div or Sub).