No, the joke is that John has been spending so much time aggressively pounding her in bed that Odie is getting jealous.
It's not saying that he wants John sexually, it's just that Odie wants some attention from his owner. Garfeild, a cat, doesn't need that attention, so it's a non issue for him.
Liz can't help but be smug, because she is getting "a bone", with john's wild, animalistic passion not just in bed, but whenever and wherever they feel the urge.
Frankly I never had any missing bits in any products I've bought.
I expect this to be relevant to 30 years ago, but in recent years, quality control for "requires some assembly" products has been very high, at least from my experience.