Students need to be warned not to play all the same moves as their opponent. The opponent has a great advantage in that they know what you're going to do! You lose the initiative in your game, then. You allow the other player to lead you, literally.
The best you could usually hope for is a draw, but many more times the result will be a trap, resulting in your loss. Why not play to win?
Copy Cat Games are formally called Symmetrical Games. Karel Traxler, Josef Krejcik, Sam Loyd and others have composed chess lines on this motif. These contrived games commonly end in checkmate.
The Batsford Book of Chess Records (2005), page 57, was my first exposure to the Rotlewi vs Eliashov sic game. It says that "the opponents were probably not playing seriously, but amusing themselves and their honorable spectators. It was the last round of the amateur tournament in the international Chigorin Memorial congress, and this draw assured Rotlewi of second place behind Alekhine, while Eliashov at best would take undivided fourth place instead of sharing 4th-6th." The commander of the Black chessmen has also been seen as Eljaschoff, for the perusers of chess databases.
Further, the Batsford Book of Chess Records, on page 57-58, claims that the game by the two masters Efim Stoliar and Januza Szukszta in Bulgaria in 1969 is the longest Copy Cat game. (This game is not at Chessgame.com, yet.) It diverges from being a true symmetrical game slightly, but it also ends in a draw.
There is a fifteen-move 2003 game between Konstantin Landa and Dierk Seifert that is a decent example of a symmetrical game-- it breaks symmetry a couple of times. It ends with resignation at apparent disadvantage of a Pawn, but a game analysis engine shows a four-point disadvantage.
Do not act like a blind, brainless imitator. Creative effort is valuable in chess. Always play with a sense of purpose and think ahead. You make progress by giving your best to each game-- by applying as many excellent principles as you have learned. Chess is not for the lazy.
This game collection is a project in progress. Games are listed in date order, not in order of importance.