(I suppose DI means Dependency injection)

No it's not a singleton, there are (<10%) cases where I need more than one object.

Otherwise I'd use a normal module.

That's why I need a mechanism for the remaining 90% to use a normal Class like a singleton module.

My interpretation:

From a Java OOP point of view Perl's module/package architecture could be understood as Singleton classes and the import routine as the constructor and package vars as class/instance variables (singleton means class=instance)

Lexical scoping of a modules effects are crucial to avoid global conflicts.

These things are hard to discuss because of little incompatibilities between languages.