Recurrence relation and time complexity of recursive factorial
Your understanding of how recursive code maps to a recurrence is flawed, and hence the recurrence you've written is "the cost of T(n) is n lots of T(n-1)", which clearly isn't the case in the recursion. There's a single recursive call, and a multiplication of the result.

Assuming the numbers aren't large, the multiplication is constant time, and it's a single operation, not a multiplier on the number of recursive calls:

$$
T(n) = T(n-1) + c_{1}
$$

The base case is $T(1) = c_{2}$

If we simplify things further, and assume $c_{1} = c_{2} = 1$ (it makes no difference to the final asymptotic analysis anyway), you get:

$$
T(n) = T(n-1) + 1 = T(n-2) + 1 + 1 = T(n-2) + 2 ...
$$

And the same final result as the StackOverflow question of $T(n) = n$.