clamp n.

late 14c., "soil of the earth; cultivated land;" also "a piece of land forming part of a clergyman's benefice," from Old French glebe, from Latin gleba, glaeba "clod, lump of earth," possibly from a PIE *glem- or *glom-, which might mean "contain, embrace" or "ball," or might be two different roots. Possible cognates include Old English clamm "a tie, fetter;" Old High German klamma "trap, gorge;" Old Irish glomar "gag, curb;" Latin globus "sphere," gleba, glaeba "clod, lump of earth;" Old English clyppan "to embrace;" Lithuanian glėbys "armful," globti "to embrace, support."

bivalve mollusk, c. 1500 (in clam-shell), originally Scottish, apparently a particular use of Middle English clam "pincers, vice, clamp" (late 14c.), from Old English clamm "bond, fetter, grip, grasp," from Proto-Germanic *klam- "to press or squeeze together" (source also of Old High German klamma "cramp, fetter, constriction," German Klamm "a constriction"), possibly from a PIE *glem- or *glom- "contain, embrace" (see glebe).

If this is right then the original reference is to the shell. Clam-chowder attested from 1822. To be happy as a clam is from 1833, but the earliest uses do not elaborate on the notion behind it, unless it be self-containment.

also clampdown, 1940 in the figurative sense "a firm, oppressive or harsh suppression or preventive action," from verbal phrase clamp down "use pressure to keep down" (1924). The verbal phrase in the figurative sense is recorded from 1941. See clamp (v.) + down (adv.).