We can say:
I have done it today
though the action itself coulp happen in the morning but we cover it with a longer peiod which hasn't ended yet.
If it happened in the morning but now it's already evening, we use Past Simple because the period itself is already over (the morning is over):
I did it in the morning
But what happens if we combine them altogether?:
I did it today in the morning
I have done it today in the morning