Do I have to use a special NAS-specific OS to make use of the NAS hardware? Like to do snapshots and stuff?
No, these features are provided by various components, which are available in any modern OS. Snapshots for example can be provided by LVM or ZFS. Disk fault tolerance (RAID) is typically provided by LVM-RAID, ZFS, or plain old mdadm, or a hardware RAID card.
Kinda related: what if I install something like Debian/Ubuntu on it? Can I still use the NAS hardware in the same way?
You can, provided you set up these components yourself. Pre-made NAS OS like OpenMediaVault or TrueNAS will have these set up out-of-the-box. Web-based configuration interfaces are often specific to these pre-made distributions, so if a Web UI is a must-have, you will have to find suitable alternatives (for example cockpit, web-based file managers, web-based user management tools, etc)