Is the saying "na'eeman" (blessings) to one who has bathed or shaved, and the reply to it "Allah yun'im 'alayk" (may Allah bestow blessings upon you), an innovation (bid'ah); given that it is a supplication (du'a) adhered to on a specific occasion?

The saying "Na'iman" (may you be blessed with comfort/bliss) to someone who has taken a bath or trimmed their hair has no basis in Islamic law. However, it is permissible as a customary practice if it is not believed to be a Sunnah, and the default ruling for customs is permissibility. Al-Nawawi mentioned that "Tabba hammamuk" (may your bath be pleasant) has no basis, and that Ali (may Allah be pleased with him) said: "You have purified yourself, so do not become impure again." Ibn Uthaymeen clarified that greetings are often a matter of customs, not acts of worship, and that they are permissible unless proven by evidence to be an act of worship. The summary is that the matter is broad, provided that it is not attributed to the Sunnah.