Is it permissible for the daughters of the paternal aunt—whom the father raised and who considered him as their own father—to embrace him, or must they observe hijab in front of him?

The children of a maternal aunt (khala) are not considered mahrams (unmarriageable kin), and their upbringing with you does not change the ruling of the Sharia that they are non-mahrams. Therefore, it is not permissible for your father's maternal aunt's daughters to appear before him without hijab (head covering) or to embrace him. Similarly, it is not permissible for you to appear before your father's maternal aunt's sons without hijab. The absence of mahramiyya (being unmarriageable kin) does not prevent kindness and maintaining kinship ties (silat ar-rahim), which are religiously required through asking about them and exchanging greetings, while observing the limits of the Sharia. Completely abandoning kinship ties may fall under the category of religiously forbidden abandonment (hajr).