We got a new machine with a SS tub back about 2014. It didn't bother Lurch in the least but he was in the kitchen laundry nook by the door with it either. I think as long as it isn't putting out any serious smells you should be fine but I'd be worried about him getting into any leftover soap spilled around the outside if he's free flying around and can get to it easily. Dobby will chew anything he can and for some reason is fascinated with soap and soap boxes so we try to keep things like that covered or put away and keep an eye on him.
Maytag, Whirlpool, some of the earlier Kenmore's, are all made by Whirlpool and there might be a few other brands not that it really makes much of a difference.