early 14c., allouen, "to commend, praise; approve of, be pleased with; appreciate the value of;" also, "take into account or give credit for," also, in law and philosophy, "recognize, admit as valid" (a privilege, an excuse, a statement, etc.). From late 14c. as "sanction or permit; condone;" in business use from early 15c.
The Middle English word is from Anglo-French alouer, Old French aloer, alloiier (13c.) "place, situate, arrange; allot, apportion, bestow, assign," from Latin allocare "allocate" (see allocate). This word in Old French was confused and ultimately merged with aloer; alloer "to praise, commend, approve," from Latin allaudare, adlaudare, compound of ad "to" (see ad-) + laudare "to praise" (see laud).
Between the two primary significations there naturally arose a variety of uses blending them in the general idea of assign with approval, grant, concede a thing claimed or urged, admit a thing offered, permit, etc., etc. [OED].
From the first word came the sense preserved in allowance "money granted;" from the second came allowance "permission based on approval." Meaning "assert, say," 19c. U.S. colloquial, also was in English dialect and goes back to 1570s. Related: Allowed; allowing.
Dutch luit, German Laute, Danish luth are from Romanic. A player is a luter (Middle English), a lutist (1620s) or a lutanist (c. 1600, from Medieval Latin lutanista).