In the Catholic tradition, are penances after confession predefined? For example would lying equate to a standard set of penances that are applied universally by all priests, or are penances set at the discretion of the priest who is hearing confession?
Penances are set at the discretion of the priest. I think there are some very general guidelines, to prevent penances that are wildly disproportionate to the sins, but there's plenty of room for discretion within the guidelines. One could get different penances for the same sins, not only from different priests, but even from the same priest on different occasions.
The priest can take into account not only the sins that one confesses but also other information that he happens to know, e.g., has this person confessed essentially the same sins to me before? He can also make the penance fit the time of the confession, e.g., giving a different penance during Lent than at Easter or Christmas, or even adapting the penance to a particular feast.
In addition to Andreas Blass' answer, in Catholicism a priest is also supposed to make penance for the sake of the penitent.
He must pray and do penance for his penitent, entrusting him to the Lord's mercy. (CCC 1466)
A priest could therefore also consider this and be willing to do majority of the penance for the sake of the penitent and give the penitent a "milder" penance.