This is the house that/where I lived in for many years as a child.
Hello!
In a test question the key says "that".
"This is the house that/where I lived in for many years as a child."
This must be a mistake, isn't it?
If it's not a mistake could you explain to me why I can't say 'where'.
Thank you!

Question

Answer

You can say
This is the house that I lived in
This is the house where I lived

You would not say “this is the house where I lived in.”

Answer

Underlying statement:  
I lived in this house for many years as a child.
"This house" is the object of the preposition "in."
"In this house" is an prepositional phrase which acts as an adverb of location for the verb "lived."

Cleft statement:  
This is the house that I lived in for many years as a child.
"That" is a relative pronoun for "the house" and is the object of the preposition "in."
"the house that ... in" is a prepositional phrase that functions as an adverb of location for the verb "lived."

Alternate statement:
I lived here for many years as a child.
"Here" is an adverb of location for the verb "lived."

Alternate cleft statements:
1 - Here is where I lived for many years as a child.
"Where" is a pronoun for "here" which is an adverb of location for the verb "lived."

2 - Here is / This is the house where I lived for many years as a child.
"The house where" acts as an adverb of location for the verb "lived."

Answer

If you are talking about location you can use where, but if you are referring to the house as a thing you would use which or that.

Answer

правильно that потому что  lived in
если бы было просто lived тогда where