Which one is correct?
He engraved photos of himself and his wife on the coins.
He engraved photos of him and his wife on the coins.
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Either is ok, and you would certainly hear both versions in conversation.
Reflexive pronouns like 'himself' are required when the subject and the object are the same person. But that is not the case here.
So, you would have to say:
He marked himself and his wife with an indelible pen.
As @aml mentioned in a comment, my first answer was in fact incorrect. Which just goes to show that a lot of native speakers don't know the rules!
Reflexive pronouns need to be used with objects and indirect objects, and circumstances such as this when "himself and his wife" are objects of a preposition, while 'he' is still subject of the sentence.
It is quite common for people to use the reflexive pronoun, either because they are not sure which pronoun to use ('he' or 'him, and especially 'I' and 'me', which many native speakers struggle with*), or to avoid repeating the pronoun, or to place emphasis on the pronoun.
*Including me apparently.