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Caption over picture of Owen and his dad..."Owen and I" is this correct grammatically ?
The difference between "Owen and I" and "Owen and me" is completely dependent on the sentence in which it is found.
If "Owen and I" are the subjects of the sentence (e.g. Owen and I are in this picture), it would be "Owen and I". If it is a picture of "Owen and me," where "Owen and me" functions as the object, then "me."
The simple test for determining correctness is to simplify the phrase to "me" or "I" (remove "Owen") and look at its correctness.
In the case of a caption, you would need to answer what the full sentence is.
Admittedly, I would rarely have a caption that just said "I", although I would say "I am doing X" in the caption. If the picture is of "me" and I just wanted to say "me" that is the more logical construction - assuming you are thinking of it in that fashion. Since I hate being the subject of my own pictures (let alone even being in them), I tend towards the implied passive "[This is a picture of ... ] Me"
That said, you really are better off with a full sentence (Owen and I are...) or, better yet, make the caption more third-person (Owen and Rita).