Animal shelter with dog inside house (7)
I believe the answer is:
cottage
'house' is the definition.
(I know that cottage is a type of house)
'animal shelter with dog' is the wordplay.
'animal shelter' becomes 'cote' (I've seen this in another clue).
'with' indicates putting letters inside.
'dog' becomes 'tag' (synonyms).
'cote' enclosing 'tag' is 'COTTAGE'.
'inside' is the link.
(Other definitions for cottage that I've seen before include "Rural house" , "Small place" , "Country residence" , "To make a home" , "Cheese pie industries all may chase this" .)