How did they conclude dolphins and whales are self-aware comparatively to humans? That seems like quite a large claim to make and the argument significantly hinges on this.
I wasn't aware of any scientific method to prove self awareness.
Note that such tests, especially the mirror test, aren't bijective. If you pass the mirror test, we can be reasonably sure you're self-aware. However, if you don't pass the mirror test (which I didn't as a toddler), that doesn't mean you can't be self-aware.
This only applies to one (quite narrow and literal) definition of self-awareness, mind you, but it's something you only see in apes and dolphins.
The article does allude to a couple of experiments in the article ("pass" option if they can't answer a question, self-recognition in a mirror). I agree it needs to be studied in more depth to make these assumptions and certainly to confer new rights.