Rachel Zegler has even drawn her own similarities between the two saying that Lucy Gray is a performer who is forced to fight, while Katniss is a fighter who is being forced to perform.
The moment itself was actually improvised between Rachel and director Francis Lawrence, who also directed Catching Fire and the Mockingjay movies. He told BuzzFeed, “Katniss had that sort of curtsy bow at the Reaping, and it was not written in the book or the script that Lucy Gray does this. I came up with the idea with Rachel for her to do it on the day. I just thought that’s a kind of cool idea to think that Katniss had heard through generations that there was a girl in the 10th Hunger Games that could’ve done this thing, and she’s doing something she’s heard about.”
Source: https://www.buzzfeed.com/laurengarafano/ballad-of-songbirds-snakes-hunger-games-easter-eggs
