I have done this with a Trotec Speedy 400 and the results were great! Although, that machine is completely overkill for etching solder mask (120W), I had the power set to 10% and the speed quite high. The settings used are similar to that of cutting paper, so just about any laser cutter should work.
You cannot mark FR4 directly, you need solder mask over copper. FR4 burns and creates nasty fumes. I've personally tried green and white solder mask, and they both turned out great. One thing, the text will be "raw copper" so this will oxidize and tarnish over time, but the readability against solder mask is good still.
Resolution fantastic, is as good as your laser cutter can achieve. Ours has 1000 DPI, so better than your average laser printer.
For marking the same thing on a batch of boards, this would be great. You size your artwork to the PCB, fill a "plate" full of the artwork in the software, arrange your PCBs on the plate in the same array, start the job. Individualised serial numbers were a pain with our laser cutter software, we had manually update the artwork SVG and re-setup the job each time.
For examples, the white one was one I've done, the green is from a RPI Zero 2 which has a lasered QR code on the back.