Some words make sense, and others do not.
Seahorse, of course, is the common name given to the genus Hippocampus – from the Greek ‘hippos’ for horse but also ‘kampos’ for sea monster. Most people would not find seahorses monstrous, but there you go. The source is mythological, from the legendary hippocamp, a sea-creature known to the Phoenicians, Greeks and to the Etruscans. Images of the hippocamp appear in ancient Greek mosaics and even on Pictish stones in Scotland. The word is a substantive for a creature of myth, later adapted to tiny marine creatures that seem to resemble coiled horses.
A sawbuck is technically the term for a particular kind of sawhorse (and thus the elision, in my mind, with seahorse), with crossed legs at each end – a kind of cradle sawhorse. It is, therefore, related to the subject of horses by several degrees of separation. The Online Etymology Dictionary (which people should look at once a day, purely because it is so lovely) says that ‘horse’ has been used since the fourteenth century for mechanical devices that resemble horses.
But ‘sawbuck’ as a slang term for a ten-dollar bill, another 19th-century Americanism, is harder to trace, since sawhorses would appear to have nothing to do with currency. HOWEVER, the etymological roots are evidently purely based on appearances – from ‘X’, the Roman numeral for ten, which looks like a sawhorse!