What does a safe do?

A safe works just like a piggy bank: it offers storage space that is tied to the player. This has two implications:

• Any safe will have the same content on any world. It's possible to deposit an item in a safe on one world, and retrieve it from a safe on a different world
• It's impossible for other players to access the contents of your safe. If they were to open a safe, they would see the contents of their own safe.

The Safe gives you more space which other players cannot access and can be accessed by you across worlds, similar to piggy bank. This can also be further increased by the Defender's Forge which is available from the Terraria-Dungeon Defender cross over.

But note one thing important that the Safe and Defender's Forge can be placed on normal ground which a piggy bank requires a platform/surface, and it is also smaller and cheaper than a Defender's Forge which makes it more convenient to use a Safe most of the time.

And AFAIK, most players end up with all three personal storage items, though some might not bring a copy everywhere they go, it is still useful to have a few lying around in the base near Crafting stations since all of the same type access one inventory which makes crafting across multiple station slightly more convenient.