rubber-stamp n.

1530s, "thing that rubs" (a brush, cloth, etc.), agent noun from rub (v.). By c. 1600 as "one who applies friction or massage in some process."

The meaning "elastic substance from tropical plants" is recorded by 1788, short for India rubber. Earlier known also as catouchou, caoutchouc, it was  introduced to Europe 1744 by Charles Marie de la Condamine, so called because it originally was used to erase pencil marks from paper, etc. Later extended to synthetic substances having the same qualities.

This substance is very useful in drawing, &c., for erasing the strokes of black lead pencils, and is popularly called rubber, and lead-eater. [from the entry for Caoutchouc in George Selby Howard, "New Royal Cyclopaedia," 1788]

The meaning "an overshoe made of rubber" is 1842, American English; slang sense of "contraceptive sheath, condom" is by 1930s. As an adjective by 1844, "In very common use from about 1875" [OED]. Some figurative phrases are from the notion of rubber automobile tires.

Rubber cement "adhesive compound containing rubber" is attested from 1856 (from 1823 as India-rubber cement). Rubber check (one that "bounces") is from 1927. The decorative household rubber plant is so called by 1876 (earlier India-rubber plant, by 1805). Rubber-chicken circuit "after-dinner speaking tour" is by 1959, in reference to the likely quality of the food.

mid-15c., "instrument for crushing, stamping tool," from stamp (v.). Especially "instrument for making impressions" (1570s). The meaning "downward thrust or blow with the foot, act of stamping" is from 1580s.

The sense of "official mark or imprint" (to certify that duty has been paid on what has been printed or written) dates from 1540s; it was transferred 1837 to designed, pre-printed adhesive labels issued by governments to serve the same purpose as impressed stamps. U.S. postage stamps were issued under the Post Office Act of March 3, 1847; Britain had them earlier, under the Postage Act of 1839. German Stempel "rubber stamp, brand, postmark" represents a diminutive form. Stamp-collecting is by 1862 (compare philately).