Does "make offline" option in Google Maps works with applications that utilize Google Maps? In newer Google Maps I can make area available offline.

Do other applications that rely on Google Maps (utilize Google Maps Android API) use offline maps?
No.
I have made my current city offline. But, when I do local search with third-party apps, the map is displayed after consuming internet connection.

Its task of third-party apps to cache fetched map data for future use.