Going more by the pictures than the translation, I think this was originally intended as an homage to the grief that is involved with having to euthanize a seriously ill pet, in context so that the owner would be feeling the cat saying it was the right thing to do, it is okay, that hey man it's cool, I still love you (from the beyond or w/e).
The last three scenes or frames of the cartoon make that very clear, at least to me. Most especially the last two frames - see the diminishment of size and how now the focus is pretty clearly about loss. It symbolizes the devastation of the moment you realize in your gut that your furry kid is really gone now, and how it makes you feel very small within that miasma of loss. "Please forget me, too...okay?" means, "Please don't grieve. It's okay. I'm free now" (from pain/illness). The English translation into vernacular meaning could have used some tweaking, is all. We all get exasperated with our pet's behaviors from time to time, and unfortunately, that is often the first thing that comes to mind right after that beloved pet's death. You feel guilty about losing your temper at them. It is the worst phase of the grieving process. I think this cartoon was focusing on that aspect.
