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Consider the Sumo Association as a corporation, which has 105 shares. (Each of them comes with a custom name.) Those shares however can only be acquired by ex-osumosan whose careers were noticeable enough (san'yaku rank, or 20+ tournaments in makuuchi, or 30+ tournaments in juyro or makuuchi.) Once you pass those conditions, you can trade, borrow, etc. those.
It is translated as "stock" simply because the word, цак, is also used for the capitalist meaning. But the word can also mean a tree stump. Perhaps "stake" would be the most fitting translation, conferring both the literal wooden meaning and the concept of holding stake. (Though "stock" also seems to have had the original meaning of a tree stump.)