Short answer is we don't know if it is the same photon or not, because our useful models treat them as fields and waves for the most part. We treat light as a particle when it's interacting with matter, but not when it's propagating. Whether we'd find the same unique ID or not is something yet to be seen maybe one day.

>Are all photons at the same wavelength identical so that it just doesn't make any sense to ask this question or are there some properties that are effectively randomized each time it is re-emitted?

This depends on the laser, and I don't think it's wise to write a whole wall of text to explain the ways in which a laser may emit photons with the same energy, but other factors may be different and make the laser better or worse for the intended application. Given the nature of lasers (and any radiating black body) we can't produce a laser that emits photons that all have the same energy; we can roughly select for a desired wavelength and it's far more efficient than with conventional light sources, but still we have for every laser emission a bandwidth. Depending on whether you want a light pulse with extreme short time duration or extremely high temporal coherence, you'll pick broad bandwidth or narrow bandwidth respectively. Unfortunately, as of now this is not a simple switch we flip, it requires the addition or removal of many optical components to the point that it's a total overhaul of the system.