First of all : Welcome to OSM. 

Summary : there likely isn't a single tag to cover this centre in its entirety: Education and Living (both on-site and at home) are recurring keywords, together with Care and Expertise. 

Then : 
for 'medical institution' you could look at the Wiki: 
https://wiki.openstreetmap.org/wiki/Key:healthcare ; 
for residential could look at https://en.wikipedia.org/wiki/Residential_area ; 
from https://wiki.openstreetmap.org/wiki/Tag:landuse%3Dresidential : "In many areas, apartment complexes are mapped as landuse=residential, residential=apartments. This can be combined with name=* for named apartment complexes, and building=apartments for the actual apartment building itself." 

Residential as distinct from Industrial or Commercial. 

But perhaps most importantly : in computing things are either true or they are false, in the real world there are overlapping functions : how you map those in very much a matter of judgement; 

'day_care': https://wiki.openstreetmap.org/wiki/Tag%3Asocial_facility%3Dday_care specifies no overnight stay.

The first couple of words on www.ganspoel.be likely summarise their purpose: "Oog voor
leren en leven , Centrum Ganspoel is een veilige leer- en leefomgeving voor kinderen, jongeren en volwassenen met een visuele en/of meervoudige beperking."

