Money in circulation, and the word really is water language.
Documented
Money as flow, documented
Currency descends from Latin currere, to run, the same root as current, courier, and course. English built its entire money vocabulary from water: banks from riverbanks, liquidity, cash flow, frozen assets, floating a loan, drowning in debt. The linguistics is real and uncontested.
Assessment: Documented etymology. The movement did not invent the water metaphors; it found them. What it did with them is the next two tiers.
Stories
The river-bank theology of money
The movement's story: because money is water-language, the banks control the current, you are sailed down it at birth, and every financial term is a hidden admission that admiralty rules your wallet. Mer-chants sell mer-chandise on the see of commerce.
Assessment: The wordplay is documented movement literature. As linguistics it confuses metaphor with jurisdiction: English describes money as water because flow is a good metaphor for circulation, not because a court hid a confession in the dictionary.
Interesting claims
The metaphor is the trap
The deepest version: it does not matter whether the water language was intended as law, because language programs thought, and a population that pictures money as a river will accept banks as its natural geography.
Assessment: A respectable Whorfian argument wearing a sovereign-citizen coat. Language shaping economic intuition is a serious research area; hidden admiralty confessions in everyday nouns is not. The archive shelves both, in different drawers.
Sources and further reading
Oxford English Dictionary, currency, current, bank
Lakoff and Johnson, Metaphors We Live By, on money-as-flow language
Boyle, The Public Domain, on linguistic capture arguments
Worth asking
If the dictionary really were a confession, who wrote it, and why did they sign it so publicly?