The phrase "close bond" refers to a strong and deep connection between two or more people. It means being emotionally and closely connected to someone. Full definition
Some dogs, especially those with very close bonds with their owners, can become depressed while in boarding or while their owners are away. (petpeoplesplace.com)
The breed is fairly active and they form close bonds with family. (pawster.com)
History is full of close bonds between people and pets. (independent.co.uk)