Reverse geocode to full address

The answer is Nominatim.

Nominatim builds an address hierarchy using information from the element's address tags and associatedStreet relations, places in the element's proximity, administrative boundaries enclosing the element and so on. Some of these steps are explained at the development overview.

If you open your link and click on details you can see the address hierarchy Nominatim has built and which other OSM elements were used to create this hierarchy.

Also note that there are some other geocoders for OSM.

SELECT string_agg( place,',') as place FROM (
SELECT name AS PLACE, SUBSTRING(planet_osm_polygon.admin_level FROM '[0-9]{1,2}')::INT AS admin_level_int
FROM
planet_osm_polygon
WHERE
boundary = 'administrative' AND
ST_Intersects('SRID=4326;POINT(78.44131 17.51671)'::geometry, way)
ORDER BY admin_level_int desc
) AS geocoder

output: Ward 129 Suraram, Greater Hyderabad Municipal Corporation North Zone, Hyderabad, Medchal-Malkajgiri District, Telangana
if you get mixed SRID geometries error change 'way' as ST_Transform(way,4326)