What is the ruling on shaking hands with worshippers and saying "May Allah accept [it]" after finishing the prayer and remembrance, and is that considered an innovation (bid'ah)?

There is no harm in shaking hands and spreading salutations after prayer, as long as it is not believed that this is among the Sunnah (definition) of prayer or related to it, because the general principle is the permissibility of shaking hands at all times. Advice and enjoining good should be done with calmness, gentleness, softness, and good speech, especially in the mosque and among worshippers, in accordance with the Almighty's saying: "Invite to the way of your Lord with wisdom and good instruction, and argue with them in a way that is best."