Pressing ENTER and using mouse to click is identical. Both produce the same result.
In which part did you try to advance? (If you mean the chibi animation and such, well, they can't be skipped with either ENTER / mouse).
Ah, it's actually advancing. Take a look at the dots "."
If you're pressing enter, the dots will increase. When it's 3 dots it will advance to next textbox. Or easier to check, if there's a mic image at the end of the sentence, that means pressing enter will advance to next textbox. If there isn't a mic image at the end of sentence, it means that sentence is not finished yet and pressing enter will show the next text (in this case, only a dot appears so it appears to not be advancing...)
But anyway, it seems to be causing misunderstanding, so I'll probably just remove those 3 dots thingy lol.