The situation is this: three Jewish lads have been thrown by Nebuchadnezzar, king of Babylon, into a furnace for failing to obey him. They are far from Jerusalem, and so cannot offer sacrifice in the temple. Moreover, they have no rams, bulls or lambs to offer. so one of them, Azariah, prays that, in place of a normal sacrifice, God will accept them themselves. The whole passage is well worth reading.
On the lips of the celebrant of a Christian Eucharist, then, Azariah’s prayer indicates that he is offering himself along with bread and wine.
The people too offer the sacrifice of themselves, and each person’s sacrifice is different : one has been working hard, another bearing sickness, another looking after children and so on ad infinitum. That is why he speaks of ‘my sacrifice and yours’.
In their reply, they speak of ‘the sacrifice’ in the singular because, as VII said in Presbyterorum Ordinis 2 (also worth a read) ‘by the ministry of presbyters, the spiritual sacrifice of the faithful is completed in union with the sacrifice of Christ, the one Mediator’. That is, at Mass all our sacrifices become one.