I'd use this sentence if the morning is still going on. Because present perfect refers to recent past.
I have had my breakfast at 6:30 today. ( in this case I use the present perfect because I consider that the day (today) is still going on in spite of the action "having breakfast" be done. )
I'd use this sentence in the afternoon or at night.
I had my breakfast at 6:30 today. ( in this case I use the past tense because the action of having breakfast is finished even though I use "today" to finish the sentence.)