If you look at some street photos (or look into official government database), you’ll see that each carriageway does have the name signed in each direction, so yeah, it’s their name. Can several related (of even disjunctive) streets/carriageways have the same name? In Croatia then can.
But for sidewalks to have names is totally weird idea over here. YMMV, of course.
For example this Way: Selska cesta (197452781) | OpenStreetMap and Way: Selska cesta (78500209) | OpenStreetMap and Way: Selska cesta (78360670) | OpenStreetMap are several related but different roads (of different classifications even!) which are all called “Selska cesta”. Is it strange? To many perhaps; in Croatia however, that is just normal (and even expected of bigger roads in the cities).
Also, the same road (with or without separated carriageways, e.g. Way: Ulica ludbreških branitelja Vukovara - Vinogradska ulica (1026218840) | OpenStreetMap) in Croatia may have one name on one side (direction) of the road, and other name on other side (direction) of the road. There is even few dozen thousands of uses for such “strange” de facto tagging in other parts of the world too.
Strange? To some, sure! To me, that is the most fun part of OSM – seeing how your expectations of how world works gets ground to dust! 
To Croatian, idea of naming a sidewalk is like you’d name the grass strip or barrier=guard_rail between those carriageways with the same name=* as the street, or naming zebra crossings over the street. (Yet I wouldn’t be too surprised to find out that to some places, it might be not just fine but expected. 
)
I guess each country have their own peculiarities.
I am no authority there (i.e. just my random personal opinion), and it might have somewhat different definition in sociological studies, but I’ll take one from basic statistics - where I’ll guess it has to do with extreme tail ends of Gaussian/Normal curve (i.e. if something is practically never done in certain group, and then some random individual does it, it is considered “weird” - i.e. as in “strikingly odd or unusual” definition, and not ones related to destiny or witchcraft; now those are some weird definitions of the word “weird”)
Good for them 
For Croatia however it does not really work in many cases, as we have this thing of changing word endings depending on how it is used, so you cannot really just drop the noun and leave the adjective.
Shortening “Ulica Jasena” to just “Jasena” would sound kind of like shorting “The Golden Gate Bridge” to just “The Golden”, but way worse. You might try to change “Ulica Jasena” to “Jasenovi” to at least try to make the word intelligible in Croatian [1], but that (apart from not being the actual name of the street) changes both the meaning and the expression and also makes it impossible to find by the name one would search for.
But I think we’re departing from main carriageway here (to another one which has the same name, but different classification and direction, if you’ll excuse my backreference 
).
that one would be like shortening “The George Washington birthplace home” to just “George Washington” – kinda keeps some meaning, but loses a lot too. Still a quite bad solution; but in English example it is much better because at least word matching still works, which it wouldn’t in Croatian ↩︎