There are dedicated label printers, accepting a roll of label media. Labels may be pre-cut or are cut by the printer. Usually, these printers use thermo-transfer, creating very robust labels. The printers usually emulate a legacy "Zebra" printer, and/or have their own manufacturer-specific printer language. Printer drivers for CUPS usually exist, so do drivers for Windows. But most times, your application generates a print job in the printer's language, without a printer driver.

I've worked with Zebra printers on a previous job, the Zebras can easily run several kilometers of labels without showing any signs of wear. You just have to clean then every now and then (paper leaves dust behind). I think that even cheaper printers for small office / home office should be able to handle more labels than you will ever need to print. The only part that may wear out is the thermal print head, depending on the quality of the labels.

Label printers come in different sizes, the smaller ones for address labels and the like are relatively cheap compared to the big Zebras that can print A5 sized labels.

Yes, those printers do not print on A4 media. But because they don't, neither the printer nor any software have to remember which labels have been peeled off and which are still usable.

Alexander