Old English tæppe "narrow strip of cloth used for tying, measuring, etc.," of uncertain origin; perhaps [Klein] a back-formation from Latin tapete "cloth, carpet," compare also Old Frisian tapia, Middle Low German tapen "to pull, pluck, tear." The original short vowel became long in Middle English.
Adhesive tape is from 1885; also in early use sometimes friction tape. Tape recorder "device for recording sound on magnetic tape" first attested 1932; from earlier meaning "device for recording data on ticker tape" (1892), from tape in the sense of "paper strip of a printer" (1884). Tape-record (v.) is from 1950. Tape-measure is attested from 1873; tape-delay is from 1968.