Is it permissible for a Syrian university student studying abroad, whose father is able to pay university fees, to request assistance from humanitarian organizations? And would this be considered among the impermissible begging mentioned in the hadith: "No servant opens a door of begging except that Allah opens for him a door of poverty"?

The default rule for asking people (for money or help) is prohibition and censure, but it becomes permissible out of necessity or severe need.

If the study is essential for you or for earning a livelihood, and you find no alternative to asking (for help) to complete it, then it is permissible for you.

However, if it is a supplementary, elective study, or if you can postpone it without harm, then it is not permissible for you to ask people (for help).

And your father is not obligated to cover the expenses of university education.