Not efficiently. There's no "clipboard changed" notification on macOS.
Such a utility would have to run in the background and constantly query the clipboard. And since the clipboard basically holds a list of types and a way to ask the owning application for data, you'd have to keep waking up the clipboard owner, too, to ask it for data. Would you check every type it declares? Some applications support dozens of types.
That would be cool, if true. It was not clear to me from the documentation that this number is guaranteed to increment whenever the contents change.
The value is described as counting "pasteboard ownership changes", and its use is to "determine whether you still have ownership". -clearContents doesn't say anything about necessarily changing it if you were already the owner, though that does seem to be the behavior today.
I suppose this could be one of those cases where we can go by what it actually does, not just what the documentation says it does. I hate doing that, because it has a way of coming back to bite me.
Such a utility would have to run in the background and constantly query the clipboard. And since the clipboard basically holds a list of types and a way to ask the owning application for data, you'd have to keep waking up the clipboard owner, too, to ask it for data. Would you check every type it declares? Some applications support dozens of types.