Glmnet uses a version of Maximum Likelihood Estimation, and for that to work you have to assume the distribution of your data. Notice that in the case of the Logit model:
Let $X = (X_1, \dots , X_k)$, $\alpha = intercept$ and $\beta = (\beta_1, \dots , \beta_k)$:
Without intercept, your error term's distribution would be:
$$G(X \beta) \sim \Lambda(X \beta) = \frac{e^{X \beta}}{1+e^{X \beta}}$$
And your estimators would be:
$$\hat{\beta}_{MLE} = argmax \frac{1}{N} \sum (y_i log(\Lambda(X \beta) + (1-y_i)log(1- \Lambda(X \beta))$$
With intercept, it would be:
$$G(X \beta + \beta_0 \alpha) \sim \Lambda(X \beta + \beta_0 \alpha) = \frac{e^{X \beta + \beta_0 \alpha}}{1+e^{X \beta + \beta_0 \alpha}}$$
And your estimators would be:
$$\hat{\beta^*}_{MLE} = argmax \frac{1}{N} \sum (y_i log(\Lambda(X \beta + \beta_0 \alpha) + (1-y_i)log(1- \Lambda(X \beta + \beta_0 \alpha))$$
Notice that the FOCs will be nonlinear and the presence or absence of $\alpha$ will change the value of the estimated $\beta$ even for $\beta_i \neq \beta_0$.
So, there is no reason to believe that the direction would be the same.
