Am I obligated to stay with my mother while I wish to travel to Mecca to perform Umrah, knowing that my father there wants me to be with him, and my mother says that she does not prevent me from obeying Allah but prefers my staying with her, and she feels comfortable with me by her side, and I have five other siblings?

If showing filial piety to both parents becomes conflicting, the principle is to combine both. If that is not possible, then the mother's right takes precedence over the father's, according to the majority of scholars.

If you can reconcile their wishes, then do so. Otherwise, remain living with your mother, without cutting off ties with your father.

As for traveling for 'Umrah: If your mother's objection stems from fear of you not returning, then reassure her that you will come back. If it is because she needs you, then obeying her takes precedence over a voluntary 'Umrah. However, an obligatory 'Umrah takes precedence over obeying parents.