Lucy, there were certainly items which were unique to each ship or class of ship, like the very prominent central light fittings which formed a focal point in some public rooms. But most of the light fittings were mass-produced, and there wasn't much incentive to pay extra for special designs and tooling when they could buy items of equal quality 'off the peg'. Some of these items, of course, would have features specific to their use in ships, like mounting brackets on 'free-standing' cabin lamps.
Again, it costs money to have items specially marked in some permanent way. During construction, it was helpful to label components to show their intended locations, but I imagine that a marking in pencil, chalk or a dab of paint would generally suffice for that. In service, it was the small, portable items that were emblazoned with a permanent reminder of where they'd come from! But that was nearly always the name and/or logo of the shipping line rather than the name of a particular ship.