That code does not implement the Singleton pattern - it's just a static class. Singleton is different: it allows one single instance of a class which is created via a private constructor to prevent others being created. It's not the same process at all!
See here:
Java - How to Use Singleton Class? - Tutorialspoint[
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And to be honest, your whole approach is wrong: the Database class should not be talking to the user at all; it should have no idea what environment it is working in. It should report problems back to the calling method and let it do the "donkey work" of telling the user.
Plus ... connections strings aren't trivial, and are prone to change - for example, the connection string you use for development had damn well better be different from that you use in production or you are going to get some really severe problems!
Connection strings will be database engine type dependant (the Access engine string will be very different from an SQL Server string for example) so you should be retrieving that from an external config file not even trying to hard code these things.