By "individualisation" I mean the idea, that everyone can have a personal relationship with God, and that the acts of every individual have an impact on the salvation of the group.
A follow-up question would be: "Was there a similar movement in other religions? Have some religions always been individualistic?"
You are bringing up two separate points: the idea of a 'personal relationship' with God, and 'the acts of every individual have an impact on the salvation of the group'. I'm answering the first one, since even this has some changes throughout history.
This is not new, but not as acute as the mainstream Christianity in the west currently. This is the essence of Christianity even from before the first Nicene Council in 325, largely demonstrated by martyrdom before 400s, and monasticism afterwards, before the advent of religious orders in 1000s.
Martyrdom was glorified to the point of the mass being celebrated on martyrs' tombs and mentioning of their names (only a few saints are mentioned) in the Roman Canon, the primary eucharistic prayer in Rome. In such cases, these martyrs often had to face conflict with members of their own family.
After martyrdom died out because of Christianity becoming state-sanctioned religion, monasticism sprang up. And this is not a communal-monasticism type practiced in the West later on, but individual monasticism where individuals would venture out into the desert to pray, do penance, and become holy.
But even in these 'movements', individual act of holiness was never separated from the Church. One cannot become holy outside the Church. And so the communal sense of Christianity in the 'one, holy, catholic, and apostolic Church' is always there. Heretics who died outside the Church were presumed unsaved.
Rather than Renaissance, it was Protestantism which actually made more acute the paradigm of 'individual and personal relationship with God'. Of course whether Renaissance partly causes Protestantism is another story. Because of the perception of the Church being a corrupt institution, salvation is now in the hands of every single person without any relation to the Church. Without any intermediaries in the form of the universal, institutional Church, salvation is relegated only to the individual.