Generally the rule is correct. We would NOT say
on this morning
on this week
on this afternoon
However, when referring to the same day in a different year/week/month, it is possible to use ‘on’ with ‘day’ (and less commonly with ‘night’). This can be used formally, and not just informally as you suggest. I suppose that originally it was ‘Upon this day…’ and the shorter form became the convention over time.
For example, today is 16th February. Did you know that on this day in 1959 Castro was sworn in as President of Cuba.
Also on this day in 1935 Sonny Bono was born. One of Bono’s most famous hits was this duet with his wife, Cher.