Kimi wa Petto (2003)

In both stories the female lead finds a male lead she knows nothing about in front of her house and takes him in as her housemate.

Loads of similarities! Both have a younger man, who do interpretive dance and is feeling lost in life, fall for an older woman, who is also in need of some company and someone to trust. The tone is very different but the characters are surprisingly similar.

Both male leads act like dogs/puppies. This one is more like Tarzan though.
Older woman/younger man age gap

Nitta Masaki is about to start work as a music teacher at a high school. The high school has experienced turbulence due to the missing person case of the prior music teacher. On the night before Nitta Masaki's first day of work, he visits the music classroom and plays the piano. There, he meets student Izumi Sae. She is his student in homeroom class. Through the missing person case, Nitta Masaki and Izumi Sae soon get closer to each other. Nitta Masaki doesn’t want to admit that his feelings for her has turned into love. Meanwhile, Izumi Sae has no doubt that her feelings for him is love. They then part ways. 5 years later, Nitta Masaki & Izumi Sae meet again. There doesn't seem to be any obstacles in front of them to prevent their love, except for one thing. That is Nitta Masaki is a married man, who has a wife.