It may seem as if we cannot know, but if we want to evaluate this then we see something different.
Dogs are mammals, just like us. Their brain structures are sufficiently similar to understand its mechanisms, such as its structures and electrical flows, the secretion of hormones related to pleasure, connection, stress, or panic.
If dogs were constantly itchy, we would’ve known ever since we started analyzing brains in terms of their structure, their function, and their interactions with bodies.
So no, dogs aren’t constantly itchy.