It's a Shia channel, I think.
Yes, it is a shia Muslim channel though this video or the topic is one of the many commonalities of shia, sunni and sufi denominations of Islam.
I have 2 questions (as a non Muslim who watched the video you shared):
1: Does a Muslim have to correct fellow Muslims or does a Muslim have to correct non-Muslims too?
2: In the end the cartoon-ish man said "If person A says that he is free to do what he wants, then a Muslim is free (or even obliged) to tell him that what he does is wrong". I like to understand this well. For example: A Buddhist says "I am free to follow Buddhism". Do you mean that a Muslim should say "you are wrong to follow Buddhism, as it is not Islam and certain things go against what is written in Koran?".
Thanks for watching and thanks for your questions.
Shia, sunni, sufi.. etc. all denominations of Islam agree upon the must and necessity of
enjoining the good and forbiddding the wrong.
Since it is mentioned in the Holy Quran;
"There has to be a nation among you summoning to the good, bidding what is right, and forbidding what is wrong. It is they who are the felicitous."
Tanzil - Quran Navigator | القرآن الكريم
also
"You are the best nation [ever] brought forth for mankind: you bid what is right and forbid what is wrong, and have faith in Allah. And if the People of the Book had believed, it would have been better for them. Among them [some] are faithful, but most of them are transgressors."
Tanzil - Quran Navigator | القرآن الكريم
And it is so important that it is one of the
Ten Ancillaries of the Faith in shia Islam.
See also Surah Tawba, verse 71:
"And (as for) the believing men and the believing women, they are guardians of each other; they enjoin good and forbid evil and keep up prayer and pay the poor-rate, and obey Allah and His Apostle; (as for) these, Allah will show mercy to them; surely Allah is Mighty, Wise."
Tanzil - Quran Navigator | القرآن الكريم
As you see this is a requirement in Islam however there are conditions and manners in enjoining the good and forbidding the evil.
As for your second question, invitation to Islam and enjoining the good and forbidding the evil is universal. But as I said, it has to be in the best manners and morals. And surely forcing is not approved. See the Quranic verse;
"
There is no compulsion in religion: rectitude has become distinct from error. So one who disavows fake deities and has faith in Allah has held fast to the firmest handle for which there is no breaking; and Allah is all-hearing, all-knowing."
Tanzil - Quran Navigator | القرآن الكريم
There are numerous books on manners and conditions of enjoining the good and forbidding the evil. Murtadha Mutahhari's book on it is the best I have ever seen: Please check
Murtadha Mutahhari
Thanks