Assume that each country deals with the registration of entry and departure of its own citizens differently.
Assume also, how some countries deal with it officially, will not always be the same as how they deal with it in reality.
Schengen Border Code Article 8 (2) defines for persons enjoying the right of free movement under Union law, what checks may be done upon entry or exit.
Schengen Border Code Article 8 (3) defines the same for third-country nationals, which includes (iii) examination of the entry and exit stamps.
Schengen Border Code Article 11 Stamping of the travel documents, define which passports are to be stamped.
With exceptions, only passports of third-country nationals, who do not enjoy the right of free movement under Union law, are stamped.
At the present time, a registration (permanent storage) of entry and departure does not take place based on Articles 8 and 11.
This will change with the intoduction of the Entry/Exit System (EES) system expected in 2022 (after the introduction of the ETIAS system), which will replace the physical passport stamp with an electronic 'stamp' contained in the EES system.
So for EU-Citizens and their long-term residents, no registration of entry and departure exists at present and in the foreseeable future.
The collection of flight data is probably the greatest source of information available for the entry and departure of persons between countries.
Nobody can claim that that this information is compleate. But it is a source of information that may not otherwise be collected. Travel information of their own citizens will be contained within this information. It is a matter who (other agencies) in the end gets this information and how it is used. Bulk collection and storage of information for later use has been known to happen in the past and who really knows what happens at present .
For the United States, this is how they enforce their exit controls for third-country nationals.
The UK also collects ferry/train information on travelers: Passport exit checks begin at UK ports and borders