If logistic regression is not linear why/how is the ordered logit regression family with GLM? I'm very confused right now. I'm studing the ordered logit and ordered probit regression and I just read that logit is not a linear model but it is logistic model but I also read that both ordered logit and ordered probit are family with Generalized Linear Models. But if logit or ordered logit is not a linear model how can it be categorized into Generalized Linear Model? Can any one explain this in an easy way to me how it makes sense? I would really appreciate if you also could give any source/reference.

GLMs do not claim to be linear!
The typical way to write a GLM uses some “link function” that I will denote with $g$.
$$
g\left(
\mathbb E
[
Y\vert X]
\right)=X\beta
$$
We can invert the (often nonlinear) $g$ and apply $g^{-1}$ to each conditional expected value given by $X\beta$.
$$
\mathbb E\left[
Y\vert X
\right]
=g^{-1}\left(
X\beta
\right)
$$
This is nonlinear, but it is nonlinear in a particular way where the conditional expected value can be transformed and then written as a linear combination of features (given in the first equation).
While this does not address your exact question, I hope this brings clarification on what is going on in a generalized linear model.
