Trademark class codes are entirely different than patent class codes. The tile of the question uses the word trademark but the question itself says you are looking into patents. Although both are handled by the USPTO, the "T" side and the "P" side of that agency are very distinct. Trademark classification in the US uses the international Nice Classification Agreement as you have linked to.
The traditional US patent classification system is called USPC. It is based on a class/subclass format and, since our system is very old we have Class 4
004 Baths, closets, sinks, and spittoons
Where we find subclass
4/ 300 Flush closet: Subject matter receptacle for receiving bowel
material and/or a liquid handling means to assist the removal of the
bowel material from the receptacle.
There is an international patent classification system (IPC) and a European classification system ECLA
Both of these are newer than the USPC. Their organization might be compared to the metric vs our ad hoc measuring systems. In ECLA-
A HUMAN NECESSITIES PERSONAL OR DOMESTIC ARTICLES
DOMESTIC ARTICLES OR APPLIANCES; COFFEE MILLS; SPICE MILLS; SUCTION
CLEANERS IN GENERAL
A47K SANITARY EQUIPMENT NOT OTHERWISE PROVIDED
FOR (connecting to water supply or waste pipe, sinks E03C;
water-closets E03D); TOILET ACCESSORIES (cosmetic equipment A45D)
A47K13/00 Seats or covers for all kinds of closets (dog or cat
toilets adapted to fit on conventional toilets A01K1/0121)
A47K13/02 • of plastic materials.
Recently the US and the EPO created yet another system the CPC. The Cooperative Patent Classification system is very similar to the ECLA but with a finer gradation of subclasses. In the last two years it seems the transition to CPC at the USPTO is happening.