The Frisch-Waugh-Lovell theorem states that the b1 vector (and e vector) in Y = X_1 * b_1 + X_2 * b_2 + e are equal to the b* vector (and e* vector) in M_2 * Y = M_2 * X_1 * b* + e*, with M_2 = I - X_2 * (X_2' * X_2)^-1 * X_2', which cleans out the effect of X_2 on Y and X_1. So in short, b_1 = b* and e = e*.
Seeing that these coefficients are unchanged in the partial and multiple regression, I wonder how it is then possible that the standard error of the b_1 coefficient is unequal to the standard error of the b* coefficient, more specifically, the standard error of b_1 (Multiple regression) is always larger than the standard error of b* (Partial regression).
If b_1 = b*, then shouldn't it always hold that their standard errors are also equal?