What is the ruling on a game that begins with a Qur'anic verse, then the participant comes up with a verse starting with the letter that the first verse ended with, or completes the first verse, with their claim that there is nothing wrong with this and it does not resemble poetry?

There is no harm in habituating young people to recall and review the Qur’an, and to alert them to the متشابه (sections with resemblances) within it. This is more befitting and beneficial than engaging in poetry. The permissibility of this is indicated by the fact that the reciters (qurra’) would stop their students at places where one would not ordinarily stop, in order to test them.