Somebody may want to take a look at the code or savestate and see if they can tell us for sure. I'm always interested on how these things are done.
Red Nintendo logo + stars:
Glowing on BG1, "Nintendo" letters on BG2, stars are sprites.
Only the letters are on both main- and subscreen, glowing is only on subscreen. The effect is that the letters stand out more through the glowing.
Fading:
Particle effect is done with ca. 80..100 sprites, each having only the top-left corner set to a visible pixel. The vertical bar is on BG1. I think the letters are removed by increasing the size of window 1 (which masks the BGs) each frame.
"Rare" logo fading in:
Logo is on BG2, sprites are used for particles.
HDMA channels are used to change the color constant (which is added) to generate the bright border, and to make the logo visible.
"Reflections" on the logo:
Mainscreen has sprites (the reflections) over the logo (BG2), subscreen (also the logo) is added to the sprites.
"Rare" logo fading out:
Mainscreen has the logo on BG2; the subscreen has a circular shaded area on BG1 that is subtracted from the BG2 pixels. The palette values for the circular area are brightened to reduce the brightness of the logo.