I don't recall ever seeing any evidence of my old MFC-8890DW being able to update itself, though just checked and see no setting for it either.
What it has instead is the PC app, ControlCenter 3, which periodically checks for updates and lets me know and have the option to download and install them. I did that once, years ago, but nothing since.
Granted I have not had a need or tried to update the CC3 app, don't know if a newer version might automate that more, but with the firmware it has currently, the printer does not appear to have its own update capability.
While I had that CC3 open, I checked for firmware updates and there are none, probably won't ever be any more for this printer that has aged out of the support cycle. I don't think the cartridges are chipped though (AFAIK), just a reset gear that indicates after so many thousands of pages (or tells the printer that a new cart was installed, I forget which) or the optical viewport on the cart to indicate low on toner.
It has used bulk toner, and 3rd party carts, and reused 3rd party carts with more bulk toner, all of it's life. It has a separate drum though, so reusing a cart a greater # of times is more viable. In my experience, 3rd party drums are usually low quality so next time it needs one, it'll be an OEM drum if they're still available.
Edit: Fast forward to Jan 2026 and I needed a new drum. Brother wanted $135!! 3rd party was $22... so I bought 3rd party, seems to work fine and if it doesn't last another 30K pages, the rest of the printer may not last that long anyway, especially by then I may no longer have any PCs running that have driver support.