You have nothing to worry about. The Mystery Gift option won’t harm your game or your DS, and it won’t mess up any features.
Mystery Gift is a feature added by the game developers themself to allow them to give players free items or Pokemon. In Generation 4, you needed Wi-Fi connection. Most events were timed, and sometimes you needed to go somewhere in person to obtain the items in the older games. Receiving these gifts did not harm the game or console at all; perfectly safe freebies!
Nintendo Wi-Fi has since been discontinued on the DS family, meaning you can no longer use Mystery Gift if you tried. They still hand out Mystery Gifts on the newer Pokemon Games for the Nintendo Switch family, though!
However, there is a DNS exploit that allows you to obtain old Mystery Gift items in Platinum. This is unofficial, and not supported through the game developers, although it can’t harm your DS. If you want to learn about how to do that in more detail, the link above is a great guide on it :)
If you want to read up more about Mystery Gifts, Bulbapedia has a whole page about it in more detail, although I’ve summarized most of it here.
Hope this helps!