Hi,
This object seems to be a building. Therefore it should be tagged as a building (e.g. building=yes, building=stable or whatever). Otherwise it will not be rendered as a building on maps.

The tag leisure=horse_riding should not be used for a specific building of a riding centre, but for the riding centre as a whole, as in 
https://www.openstreetmap.org/way/515295027#map=18/41.59546/20.67327&layers=P and https://www.openstreetmap.org/node/7310645719#map=19/42.69120/21.11865&layers=P

See also https://wiki.openstreetmap.org/wiki/How_to_map_a#Riding_centre_(equestrian)