You will get some results, for sure. Depending on your application may even be good enough. But as a general probability that an image is something other than cat and dog, not really.
As other commenters have mentioned the general problem is known as OOD (out of distribution) sample detection. There are Deep Learning models which model probabilities explicitly and can in principle used for OOD sample detection - Variational Autoencoders. The original formulation of this model performs poorly in practice at OOD sample detection, but there is work addressing some shortcomings, for instance Detecting Out-of-distribution Samples via Variational Auto-encoder with Reliable Uncertainty Estimation. But with VAEs things get very mathematical, very fast.
Coming back to you initial question, no, softmax is not appropriate for "confidence", but this is an open problem in Deep Learning.