To start, I'm sure pensiones is not the right word in Spanish since that's what we stayed in when we were in Cinque Terre a couple of years ago. If there are any in Madrid can anybody recommend a few?
They are called "hosteles" now. There are tons but I don't know any, sorry.
A pensione in Italy is a catholic religious residence. From what I can see of the hostels in Madrid seem to have a religious affiliation.
There are literally hundreds of "pensiones" (in singular, "pensión") all over Madrid (and thousands in Spain), and regulated as such. For information on the difference among types of lodging, https://gestiona.comunidad.madrid/wleg_pub/secure/normativas/contenidoNormativa.jsf?opcion=VerHtml&nmnorma=13158&eli=true#no-back-button, Google translate can help.
A pensión in Spain is normally family -run and offers basic accommodation, and may offer meals too.
As Mikel stated, there are a number of pensiones in Madrid. I doubt many have religious affiliations. Some are hostels, some guest houses, but many are pensiones. Search ”Pensión, Madrid” in Google maps and many results are returned.
The only religious associated lodging I found in Madrid is Residencia filipense "Gertrudis Castañer".
I would never associate a pensión with any religion. Not in Spain. And "hostales" are not religious, either.