The industry argues this as one question. It is really two, and the answer depends on what the part does.
For load-bearing and safety-related items — hooks, wire rope, brakes, limit switches, load cells and anything in the safety circuit — the original specification is what the crane was calculated and certified around. Material grade, tolerances and fatigue behaviour are part of the design, not details of the purchase, and a part that looks identical can change how the components around it wear. On these we supply original specification.
For commodity items — bearings, contactors, festoon trolleys, collector shoes, buffers and cable — the "original" part was bought from a component supplier in the first place. Here an equivalent part of the correct rating is the same part, and we will say so rather than sell you a badge.
Where a crane is old enough that the original item is no longer made, we say what can be supplied to the original specification and what would have to be produced to drawing, before you commit to anything.