The words lengthen and elongate are characteristic of the way the English language may have two words with virtually identical meanings: one Germanic and the other Latinate, either directly from Latin or through French.
Lengthen arose in the late 14th century from adding the factitive -en suffix often used to transform adjectives into verbs (shorten, whiten, broaden, widen). Earlier speakers had simply used the adjective as a verb: to length (to brown meat, to short an investment, to book a hotel room, to map the area).
Elongate first appears in the 1530s, as the Online Etymological Dictionary explains, from Late Latin elongatus, p.p. of elongare ‘to prolong, protract, remove to a distance’. Elongate replaced the earlier borrowing elongen
(mid-15c.), formed with the same -en suffix as lengthen.
English speakers regulary apply both verbs to the human face:
When styling a rounder face, your main goal is to elongate the face and make it appear more oval.
To slim down and lengthen the face, opt for a beard that is relatively long at the chin and shorter on the cheeks.
Despite a general preference for lengthen over elongate, a Google NGram shows that the frequency difference between the two verbs with the direct object face is now insignificant:
When the past participle is used attributively as an adjective, however, there is a strong preference for elongated:
This would suggest that the authors of the test simply didn't think of the word elongate as they were writing the question. There are, however, no grounds, either in meaning or usage, to consider elongate an error.
This does not mean that the two verbs are complete synonyms. Elongate is not often used in a temporal sense, for example, except in an elongated stay at a hotel. And a street is extended or lengthened. But with the human face, they are.