It's a very good question but I don't think it's impossible to probe what consciousness is merely because we can only directly experience our own consciousness.
There are plenty of other things that we can't directly experience but we can still use science to investigate them through indirect observations.
I think the biggest insights into consciousness so far have come from people with brain traumas and dysfunctional. E.g. people who have had their brain halves cut, or that guy who got most of his brain taken out by a bit of rail, or the Memento guy.
And it definitely is a thing that exists in our physical universe. A very weird thing, sure, but science doesn't say stuff can't be weird. We can definitely learn about it.
Maybe we'll never learn fundamentally what it is, but that's true of normal physical things like atoms and time too.