If the kitties like it it's all good.
There is a great deal of confusion about blocking. It's less about changing the size or shape of a FO than about making it look more finished and even. A lot of material on blocking was written when yarns were less sophisticated, almost always wool, and harder to work with.
In most cases, blocking a sweater just involves dampening the FO, smoothing it into shape and letting it dry on several layers of bath towels. No fancy equipment needed.
With some complicated patterns and with certain yarns, it's good to lay out the pieces before you sew them together, then dampen, pat them out to the correct measurements, and let dry. If you pat the edges down smoothly it makes sewing-up easier.
You can also improve the look of some handknit fabrics by steaming or pressing lightly. I haven't done this in years and don't intend to except under extreme duress.
Yes, you can stretch a FO by blocking, but not by much. Don't count on blocking to change a size Small into a size Large. And yes, if you stretch a sweater by blocking, you'll probably have to do it every time you wash it.
(I won't mention blocking lace here; it's a whole other thing.)
Your rug looks fine. Add the fringe, if you want to add size. Or pick up stitches all around and add a border, maybe of another color.