Storing information just so it persists between scenes does not require a database. Storing information online and sharing it with others does require at least a simple form of a database.
What you're asking is two things.
1) Storing information on the client side. You can store information in PlayerPrefs, so it will persist even when the app stops and restarts.
2) Creating a multiplayer game. This is a lot more difficult. Make sure that you are ready for a task like this. If so, consider what exactly you need from your online system, check out database systems (NoSQL vs MySQL), serialising data as JSON objects etc.
Also, for multiplayer games (depending on what platform you'd like to build) check out GameCenter, Google Play and (if not iOS / Android) Photon Multiplayer solutions.
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