Is the approval of a girl's family for a young man considered an engagement, and is it permissible for him to message her, with her family's knowledge and approval, given that the content of the messages is limited to reassurance, religious topics, and motivational discussions?

If the woman and her guardian have approved the engagement, then she is considered engaged, even if the usual procedures have not been completed. Your conversation or correspondence with her, whether she is engaged or not, does not alter the ruling, because she remains a stranger to you until the legal marriage contract is concluded. Thus, correspondence is permissible for a need, without repetition or habituation, to block the devil's pretexts. It is advisable to initiate and hasten the marriage, as it is among good deeds.