Патриарший домовый монастырь is an institution so unique that one can't draw an analogy with any outwardly similar concept, such as house church or ecclesia domestica ('domestica ecclesia' would sound illiterate in Latin).

1. Ecclesia domestica = an observant religious family
As defined and explained by Pope John Paul II himself:
Seen through Christian eyes, the family is “the domestic Church” (ecclesia domestica). The Christian family, like the Church as a whole, should be a place where the truth of the Gospel is the rule of life and the gift which the family members bring to the wider community. The family is not simply the object of the Church’s pastoral care; it is also one of the Church’s most effective agents of evangelization
http://saints.sqpn.com/pope-john-paul-ii-ecclesia-in-asia-th...
In simpler terms:
Domestic Church is the English translation of the Latin expression, Ecclesia Domestica. Strictly literal, the Latin word ecclesia is related to the English word church and the Latin word domestica is related to the English word house. But the whole expression is understood as the “Church in miniature” or “small-scale Church.” Yes, the Christian family is the “Church in miniature” or “small-scale Church” because the Church into which her mystery the Christian family is grafted, is the “large-scale Church” - the whole Church
http://en.wikipedia.org/wiki/User:Papatz
A similar explanation:
Vatican II also spoke of marriage as a microcosm of the Church, the Ecclesia domestica: "In what might be regarded as the domestic Church, the parents, by word and example, are the first heralds of faith to their children." 9 In other words, the "home is the first school of Christian life."
http://www.catholicculture.org/culture/library/view.cfm?recn...

2. What is 'house church'
In Russia, домовая церковь is an independent church intended for members of a certain household or an institution <...>, unlike a parish church intended for everyone and under jurisdiction of the corresponding administrative unit of the Russian Orthodox Church
http://en.wikipedia.org/wiki/House_church_(Russia)
In the West, 'house church' is an informal term that has a different meaning, see
http://en.wikipedia.org/wiki/House_church

It is clear from the context
"подлежал ведомству патриархов"
that the Patriarch's monastery in question
- is administered by the Patriarch as primate of the Russian Orthodox Church, and is therefore an official ecclesiastical institution
- is not analogous to a house church (which is independent, not under the jurisdiction of the Church)
- has nothing in common with the Catholic idea of ecclesia domestica (family, see above) since the Patriarch is a monk
It is a grand complex of buildings rather than a minor structure; e.g., see
http://www.solanatour.ru/rus.html