On a smooth horizontal floor, an object slides into a spring which is attached to another mass that is initially stationary. when the spring is most compressed, both objects are moving at the same speed. ignoring friction, what is conserved during this interaction?

Assuming there is frictionless motion this interaction doesn't involve any external forces .. and if the spring is 'ideal' (perfectly elastic), then there is no energy transferred to internal heating of the spring as it compresses. 
In reality the spring will transfer some energy internally to heat as is compresses .. so only under ideal conditions will all the initial KE transfer to strain energy within the spring and mechanical energy be conserved. 

Momentum is always conserved when all the bodies involved are considered .. as there is supposedly no friction, this interaction conserves momentum (there is no force exerted on an external body so no momentum is transferred out of the system) 
Momentum is transferred ('gained' or 'lost') by a force (F) acting on a body for a time (t) .. Newton law 2. 
According to Newton law 3, 'action' and 'reaction' forces are always equal and applied in opposite directions to different bodies .. and act for the same time. Because of this, momentum losses from one body equal the gains by another .. and net momentum is conserved.