During episode 22, we see a test between Sarada's DNA and Karin's preserved umbilical cord. I can understand Orochimaru being the type of person to keep DNA on all his staff, but is there any symbolism, cultural significance or plot reason why in particular an umbilical cord?
I don't think there is, my guess is that it's just a real life reference to the medical field where umbilical cords are kept, as they have stem cells that are helpful to treat some conditions.
Those cells can differentiate into different types of cells, which would give some scientific base to some actions he has done, as injecting Hashirama's cells.
That wasn't Karin's preserved umbilical cord, as Suigetsu wrongly led us to believe at first. In fact, it was the umbilical cord that belonged to Sakura and Sarada from when Sakura gave birth to Sarada, as it was explained by Karin herself in the manga "Naruto: The Seventh Hokage and the Scarlet Spring" (chapter 10)/in the anime "Boruto: Naruto Next Generations" (episode 23).
Keeping in mind that the DNA from an umbilical cord is the same as the DNA of the child, this was the reason why there was a perfect match between the DNA belonging to the umbilical cord and the DNA from the sample provided by Sarada herself.
The inclusion of the umbilical cord in the story took place mainly because there had to be something that (wrongly) made Sarada's doubts seem true for a while in order for the story to be able to move forward. Having a DNA test done in order to seemingly concretize Sarada's suspicions about her parentage for a short while and not receiving a concrete answer to any question about this subject until the very last chapter represented pretty much the only way this kind of plot could evolve.