Is it true that Haiti and Liechtenstein found out during the 1968 Olympic Games that they had the same flag?

No, the unfortunate coincidence was discovered at the 1936 Olympic Games in Berlin.

For a very long time, Haiti and Liechtenstein used the same flag design without knowing. Both countries had a bicolored flag that featured two equal-sized horizontal bands of blue and red colors at the top and bottom, respectively. It was during the 1936 Summer Olympics that Liechtenstein and Haiti discovered that they shared the same flag.

https://www.worldatlas.com/articles/which-two-countries-used-to-have-the-same-flag.html
https://www.grunge.com/281063/how-the-1936-olympics-made-two-countries-realize-they-had-the-same-flag/

Further to AyatollahK's answer, the Liechtenstein flag consisting of just blue and red stripes was made official in October 1921. Once they realised it was the same as Haiti's at the 1936 Olympics, they changed the flag on 24 June 1937 by adding a crown in the top left corner to signify the country's position as a principality, and to differentiate it from the flag of Haiti.

The national flag of Haiti was actually not just blue and red stripes, as it had a coat of arms in the center. However, the civil flag, which omits the coat of arms, was often used at events such as the Olympic Games, and so the flags at the games were indeed identical in design. The flag of Haiti has undergone two redesigns since 1936.

https://icetonline.com/two-countries-showed-up-to-1936-olympics-with-the-exact-same-flag/
https://en.wikipedia.org/wiki/Flag_of_Haiti
https://en.wikipedia.org/wiki/Flag_of_Liechtenstein