> What even means the 'public debt' number when a country have its own currency and inflation is under control?

This is a question that you have to ask when using the analogy between a country's finances and e.g. a household. It _can_ mean that one part of the country (government, central banks) has created and loaned money to _other parts of the same country_. In other words, liquidity and investment. i.e. good things! This is fundamentally unlike a household.