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Has there been a formal study done by the Church? Not necessarily. But there are plenty of books and articles and homilies and podcasts on the topic:
The short answer is yes, the name 'house of bread' has significance when relating to His other teachings. Christ was born in a manger in the town 'house of bread.' Let's break that down.
What exactly is a manger? A manger was used as a feeding trough for the swine. So the obvious significance here is that Christ was born and immediately laid in a place where food is eaten from. Think of the symbolism here: Jesus would institute the Eucharist, His own flesh and blood for us to consume, the same flesh and blood that was born into a feeding trough. And where was that feeding trough? The house of bread! His flesh would be presented in a trough for us to eat under the disguise of bread. Manger in the house of bread, Bread of the Eucharist on the altar.
Another point, 'manger' comes from a French word that means 'to eat.' When translated from Greek, John chapter 6 uses the same word when Jesus says to eat (or chew) his flesh. (https://fellowshipoftheminds.com/2012/12/30/the-meaning-of-jesus-being-born-in-a-manger/)
There is also some symbolism about Him being born into a feeding place of swine because the Jews could not eat pork, but that is for another topic...