Sick like a dying dog

I have been 'under the weather' the past 2 days, perhaps it was something I ate. It must be, since whatever that's making me ill, is not related to influenza by symptoms. Either that, or my stomach decided that it was time for acid wash and loaded me with gastric juices.
That, caused me 48 hours of awake time, since I was sleeping my discomfort off. I only woke to pop more paraceatamol, sip cold lemon tea, eat sour plum and fresh honey melon. I guess the lemon was not a good idea, since it's acidic, but hey, my tongue lost it's ability to taste and I needed to keep myself hydrated.
I was sick, and being an "up & about" person, staying on bed meant, suffering. I was suffering by normal standards, but I didn't think I was dying. So why on earth do we say 'sick like a dying dog'. People die without feeling sick, and I'm sure, if my dog woke up one morning and found herself in heaven, it was going to be because she is old and she probably went whilst napping. I wonder why not say 'sick like a dying cat' or is it because cats have 9 lives? Jeez - it must be my sick physical state, since this is really an opinion that wouldn't matter, because it's been so, for generations who have studied English Language.
BUT it simply crossed my mind, because I didn't have anything better to do the past 48 hours, except waste the minutes away. It's worth a thought though - dying doesn't mean sick and sick doesn't mean dying...DOG or otherwise!