It's confusing.. there's a bit of legacy here. As noted in the docs, there are two mechanisms for following a path.. 'Path Animation' is now deprecated, but still available.
CtrlP > 'Follow Path' sets up Path Animation in the curve, with no constraint on the object.
CtrlP > 'Path Constraint' gives you the constraint on the object, and automatically switches on 'Path Animation' on the parent curve
When you manually assign a Path Constraint, 'Path Animation' is automatically switched on in the curve you select as its target. It has to be to get movement.. but...
Checking 'Fixed Position' puts the constrained object under sole control of the 'Offset' field, ignoring the Evaluation Time in the Path animation.. and parameterizes the curve, 0-1... (and in some circumstances, you have to 'animate the curve' by hitting that button in the constraint.)
Overall impression, two systems fighting, for a while, at least.
Personally, I prefer CtrlP > 'Path Constraint', followed by 'Fixed Position', and keyframing the object's location on the curve, 0-1, in the Offset field of the constraint, by hand. That way, I don't have to dodge backwards and forwards between the curve and the constrained object.