There is no prohibition in using the older formulae.
Many people assert the new rituals as if they cancel out the older traditions, nothing could be further from the truth.
Our Church was not founded in 1965, many folks act as if it were.
I agree on not taking any cues from popular media – that would be the WORST place from which to get one’s catechesis !
While there is nothing wrong with using the older formula, and it is still commonly taught, it is true that it doesn’t fit as well in the new rite. In the old rite, “Bless me, Father, for I have sinned…” was followed by a blessing from the priest. In the new rite, the person asks for a blessing, which is not forthcoming.
As far as the original question goes, no, you don’t have to say those words in confession. It is the form commonly taught to Roman Catholics in the U.S. and many other English speaking countries, but it is far from universal or required.
In Spanish speaking countries, the priest begins “Ave Maria purisima” (Hail Mary, most pure) and the penitent replies “Concevida sin pecado” (conceived without sin)
In the Byzantine Catholic church, we begin “I confess to Almightly God, one in the Holy Trinity, to the Blessed Virgin Mary, Mother of God, and to you, Father…”
There are many variations, most of which seem language and culture specific.
The important thing is that we go to confession. A feeling of awkwardness, because we don’t think we know the right words to say, the right formula to use, can keep us from confession for years. If you get yourself to confession and ask the priest for help, he will walk you through it.