Learn English – Why does “bananas” mean “crazy”

The definition of a fishwife is tied to both of the definitions you listed. Wikipedia's article on the fishwife explains that:

A fishwife or fish fag is a woman who sells fish.... Fish women were notoriously loud and foul-mouthed as in the expression, To swear like a fishwife. One reason for their outspokenness is that their wares were highly perishable and so lost value if not sold quickly.

Unlike its common usage today, wife meant any woman, not just a married one. Some words still retain this meaning:

This usage stems from Old English wif (woman) and is akin to the German weib, also meaning "woman". This sense of the word is still used in Modern English in constructions such as midwife and old wives' tale.

So historically, a fishwife was just a woman who sold fish. Over time, since fishwives were often "loud and foul-mouthed," their job title became synonymous with your definition of "a bad-tempered woman with a loud voice."

Interestingly, fishwives have had different reputations in different areas. In Billingsgate, there were "the wives of Billingsgate" who:

dressed in strong 'stuff' gowns and quilted petticoats; their hair, caps and bonnets were flattened into one indistinguishable mass upon their heads. ... They smoked small pipes of tobacco, took snuff, drank gin and were known for their colourful language.

On the other hand, the fishwives of Newhaven, Scotland were:

noted for their beauty and industry, and celebrated by royalty.

This one’s complicated in a fun way, according to the history related by Merriam-Webster. A word for a small garden snail came to be used for small, stout-bodied men (especially in a scornful way) and for small, stout-bodied birds, and mutated considerably as it traveled over time.

In the beginning, there was dodman. We don’t know its origin, but it meant, and in some locales still means, a snail. (See more at Wikipedia.) This was altered to

hodmadod, “snail” or “snail shell”, which acquired a secondary meaning of “deformed or clumsy person” or “scarecrow”. This in turn altered to

hoddy-doddy, also meaning “garden snail” or “snail shell”, and additionally used to mean “short and stout person”, “henpecked man”, “cuckold”, and “fool, blockhead, simpleton”. Then, much like the compounds “lamebrain” and “blockhead”,

hoddypoll began to be seen. From hoddy-doddy + poll (head), it meant “fumbling inept person” or “cuckold”, but apparently also preserved the original sense of small and stout-bodied: it was shortened and altered by way of noddypoll to

noddy as early as 1530, meaning “stupid person” but also a kind of stout-bodied tern, which brings us to

tomnoddy, from Tom (nickname for Thomas) + noddy, meaning “fool, dunce, noddy” but in Scotland also a kind of Atlantic puffin. It is like calling someone Jack Pumpkinhead or Joe Cool.

When Tolkien uses it

Old Tomnoddy, all big body,
Old Tomnoddy can’t spy me!

it is clear he is using it in both its sense of “stout-bodied” as well “inept” or “stupid”.