Yes, we can on many levels. I am not sure who says these things
Never heard of the "universe started Thursday" theory?
Essentially, there is no proof that the universe didn't start last Thursday. All of your memories, your experiences, your tangible progress, could be planted and you would never know.
So how do you know you are "you" as you think you are, or if you're just a week old construct that believes you are "you"?
Also, I think that the whole "we can't be certain we are the same person who wakes up every morning" is based on the ship of theseus concept they were building on.
You wouldn't consider yourself the exact same person you were when you were 5 for obvious reasons. So it stands to reason that that change happened at some point. How would you know that you did not change over night? And if you did, are you the same person as yesterday? And if you answer yes, where's the line? Are you the same person as last year? 5 years ago? Obviously not, so how can you know that that caliber of change hasn't happened to you in a night, or that any amount of change makes you someone else?
Also, they could be referring to the broken consciousness theory, where consciousness is destroyed when you fall asleep, created when you wake, and dreams are an illusion.
In that scenario, if your stream of consciousness actually is broken, can you say you are the same person as yesterday? If the breaking of consciousness doesn't matter in that question, would a perfect copy of you with all of your memories also be you? Or not, because you can't experience their perspective?
I think the break here is whether or not you can define consciousness as "you". For your supposition to be true, the answer would necessarily have to be no, as you said you can prove that you are yourself in many other ways.
But without a point of perspective experiencing the universe, what are we?