Is it permissible to marry and bring the wife to the family home due to financial hardship, or is it better to be patient, knowing that both spouses are 19 years old and still students?

If marriage is made easy for you, then praise be to Allah. Otherwise, be patient until Allah makes your matter easy, and work on everything that helps you maintain chastity and steadfastness in obedience to Allah, such as fasting, keeping company with the righteous, occupying your free time with what is beneficial, and avoiding causes of temptation. Studying does not prevent marriage, neither Islamically nor customarily. If the girl agrees to live with your family and your family agrees to that, then there is no harm. The scholars have stated that if the father is well-off and his poor son needs to marry, it is obligatory for the father to marry him off and provide for him and his wife.