"process of shaking; downfall, overthrow," 1610s, noun of action from Latin labefactus, past participle of labefacere "to cause to totter, shake," literally and figuratively; also "to overthrow," from labare "to totter, stand unsteadily, be ready to fall, begin to sink, give way" (which is perhaps related to labi "to glide, slip, slide, sink, fall; see lapse (n.)) + facere "to make, do" (from PIE root *dhe- "to set, put").
Alternative labefactation (from Latin labefactitionem "a shaking, loosening," noun of action from past-participle stem of labefacere) is attested from 1775. As a verb, labefact is from 1540s, labefy 1620s, labefactate from 1650s.
Meanings "dangling strip of cloth or ribbon used as an ornament in dress," also "strip attached to a document to hold a seal" both are from early 15c. General meaning "tag, sticker, slip of paper" affixed to something to indicate its nature, contents, destination, etc. is from 1670s. Hence "circular piece of paper in the center of a gramophone record," containing information about the recorded music (1907), which led to the meaning "a recording company" (1947).