Not that Iason fully has a problem with that, but as Riki isn't typical pet material. It takes Iason three years to train him into one. Which seems to involve all sorts of marvelous devices like chains, drugs, pet rings, and Daryl (Iason's Furniture). Iason also breaks many rules in his relationship with Riki. He keeps Riki past the typical pet age, he also has sex with him, both acts cause his friend Raoul (another Blondie) and Jupiter to attempt to warn him. He doesn't listen to either of them though.
After the three-year training period, in somewhat the belief of "if you love something set it free," Iason lets Riki go. He doesn't take his name off the pet registry and a year later reclaims him. There's more to that phrase though, and the ending plays out in a rather sad way. The anime is highly recommended, you can find it at many fansub distributor sites.