I looking for some comprehensive database about every nature trails in countries which contains basic information about nature trail and map. For example in Slovakia there is database naucnechodniky.eu.
Open Street Map, depending on country / region / city. 'every trail' and even the word 'nature' is very tall order, trails are combined with other uses, and built and maintained by many, many different neighborhoods, cities, metropolitan areas, state / provinces, and federal governments and NGOs - official and unoficial. For instance, many 'trails' that appear as a footpath through the woods can nominally be 'roads' in the database, some natural trails turn into streets as they pass through urban ares.
So you need to become, very, very familiar with OSM's Tag Info https://taginfo.openstreetmap.org/ and one needs to really delve into how the tagging structure works in a given area, it is not like a conventional GIS with a rigorous schema, and make no assumptions about completeness - quality varies according to the strength of the local OSM community. If you notice in the graphic, what you call a 'trail' is (possibly) 'highway=path' in OSM. You can then use OverPass to extract the data, and that has another learning curve: https://taginfo.openstreetmap.org/tags
It is highly recommended you sign onto the appropriate 'local' OSM mailing lists and forums and ask questions: https://wiki.openstreetmap.org/wiki/Mailing_lists