Your "first discovered by" Tag will still be there. So if you open up the system map on a system you know you've already discovered, or if someone else visits a system where you first discovered things, they will still see this (only with your name there):
If you go back there and use the DSS to map "your" planet (and then sell the mapping data somewhere), it will look like this:
And if someone else uses the DSS and sells the data on one of "your" planets first, or if you discover and Map a planet that's been First Discovered by someone else, it will look more like this:
So, you don't "lose" any First Discovery Tags; they just get shared with whoever gets the First Mapped Tags, whether that's you or someone else.