The Jain word kashaya originally names a sticky or astringent substance, the sap of certain trees used in dyeing and tanning, and Jain philosophy adopted it deliberately: mana and the other three kashayas are what makes karmic matter cling to an otherwise unburdened soul, rather than simply passing through it as neutral action would.
Padmanabh Jaini's study of the tradition explains the mechanism in some detail: every action generates karmic particles, but only action performed with kashaya present causes those particles to bond and ripen into future consequence, which is why a liberated being, a kevalin, can still act, even eat and walk, without generating new bondage, having eliminated the passions that would make karma stick. Mana specifically is graded by intensity in Jain texts into four degrees, from a fleeting flash of pride lasting moments to a fixed egotism that persists for a lifetime and binds karma for correspondingly longer, a scheme applied to all four kashayas and used to calibrate exactly how much spiritual progress a given lapse costs. Buddhism reaches mana from a different direction and without the karmic mechanics: in the five Buddha family scheme it is read as the distortion of the wisdom of equality, associated with Ratnasambhava, the confused version of a clarity that would see all beings and all phenomena as fundamentally of equal worth, so that overcoming mana and overcoming the comparing mind are, on that account, the same task seen twice.