Is it permissible to change the password of an email account belonging to a young man who sends romantic messages to a girl, after his account has been hacked, or would that be considered an intrusion?

It is forbidden to spy on Muslims, and it is not permissible to access your brother's email. Rather, you must conceal his actions and not change his password. Instead, you should advise him and explain the impermissibility of relationships with girls. If he persists and does not desist, and there is a fear that the matter will escalate, then it is permissible to take legitimate means to change the reprehensible act, such as informing someone who has authority over them, and threatening him with that. Changing the password is permissible if it severs the relationship, but he might open another email. Use wisdom and good exhortation in changing the reprehensible act.