What is the grammatical term when the verb and the object are the same word? Examples: Plant the plant two inches deep. He dreamed a dream about Christmas.
This is called a cognate object. 'They danced a dance.' sounds ridiculous, but 'They danced the dance of death' sounds acceptable.
cognate object [grammar]
a noun functioning as the object of a verb to which it is etymologically related, as in think a thought or sing a song
In 'they danced a tango', tango is a hyponymic object.
It depends. If the verb is usually intransitive (like "dream") it is called a "cognate object construction".
If the verb can be used in a similar way as a transitive verb with other types of objects (e.g. with "plant" we can say "plant a tree" or "plant a bush") the term "cognate object construction" may not be considered to apply. But this distinction can be a bit uncertain.
In a Cognate Object Construction (CO construction, COC) a typically intransitive verb combines with an NP which has the same meaning or the same morphological stem. [...] In addition to this semantic and morphological characterization, we only speak of a COC if the verb is highly restricted with respect to the nouns that it may combine with. Consequently, in (2-a) and (2-b) we have a real CO, while in (2-c) the noun dance can be considered a regular direct object and in (2-d) the noun growth is on par with other extension NPs.
(2) a. Smith died a gruesome death/ *a murderer/ *a suicide.
b. Sam lived a happy life/ *something happy.
c. Smith danced a jolly dance/ a jig.
d. The tree grew a century’s growth/ a century’s expansion within only ten years. (Nakajima, 2006)
While the distinction between real COs and apparent COs seems important, authors differ with respect to how they classify individual examples. It seems, thus, that an adequate characterization of the COC should also provide a basis for explaining why the (2-c) and (2-d) examples are sometimes taken as COCs.
("The Family of English Cognate Object Constructions", Manfred Sailer, 2010, p. 192)
Sailer indicates that there are different types of cognate object constructions.