Both PlayStation 5 and Xbox Series X have SSD drives, which are still very expensive to buy. That's why the former only has 825GB storage to offer, with 1TB for the latter. You won't be able to add a second drive as easily next generation either, as the games are specifically made for SSDs. That's why you'll be able to change drives for PS5 while Xbox Series X has 1TB Storage Expansion Cards.
It still sounds like storage will be a problem next generation, but now Xbox director of program management Jason Ronald, has shared intel with Gamespot about a trick that Microsoft's console can do that might even make file sizes smaller compared to this generation.
Amongst other things, Xbox Series X offers a feature called Hardware Decompression (which you can read more about here), and Ronald explains that this "allows games to consume as little space as possible on the SSD while eliminating all CPU overhead typically associated with run-time decompression. This includes BCPack, which is designed specifically for compressing texture data that often is one of the largest contributors to overall game size... to help compress the actual size of the assets."
Ronald also reveals that you will be able to install certain aspects of a game, which means you won't have to install all language packages you don't need or skip single or multiplayer portions of a specific game:
"Developers can also take advantage of a platform feature to only install the pieces of the game players need, thereby minimizing the amount of content that needs to be installed or downloaded to the SSD. We're providing a lot of tools to help them be more efficient. If you're on an English console, [you shouldn't] be taking space for French or Spanish audio files, things like that."
If this is going to be enough, it remains to be seen. As games are already way above 100GB today, we are grateful there are steps being taken to reduce file sizes.