I'm learning about Use and Supply tables, and all the tutorials (including this very nice video from the Singapore Department of Statistics) show that products are on the rows and industries are on the columns.
The Use and Supply tables do not have industries on both rows and columns. For example, as shown in the screenshot below, the Supply table has commodities on the rows and the industries are the columns. Perhaps the confusion is coming from the fact that the commodities and the industries use the same encoding. For example, as you point out, why is there a commodity called "Farms" that use the code 111CA? This is because the commodity called "Farms" represent the commodities produced by the industry called "Farms."
As I mentioned, the commodity called "Farms" represents the commodities produced by the industry called "Farms." This can be problematic because different industries may produce the same good. For example, the restaurant industry produces food while the hotel industry may cook meals as well as provide rooms to stay in. This problem is handled somewhat in the "redefinitions" process. You can take a look at tables before and after redefinitions here. You can read more details in the BEA IO tables handbook "Concepts and Methods of the U.S. Input-Output Accounts".
Also, you should note that there is an effort to develop and introduce a new, separate classification system for products that is independent of the industry classification system. For example, the BEA uses a version of the North American Industry Classification System for industries and will eventually start using the North American Product Classification System (NAPCS) for products. As explained by the US Census Bureau, this has the following benefits:
The North American Product Classification System (NAPCS) is a comprehensive, market- or demand-based, hierarchical classification system for products (goods and services) that (a) is not industry-of-origin based but can be linked to the NAICS industry structure, (b) is consistent across the three North American countries, and (c) promotes improvements in the identification and classification of service products across international classification systems, such as the Central Product Classification System of the United Nations.
You can read more about the status of NAPCS here.