The Atari 8-bit computers can but only with 4 colors chosen from 128.
The Amstrad CPC computer can with 16 out of 27 colors. The Amstrad CPC Plus models can with 16 out of 4096 colors.
I believe the MSX 2 and 2+ models can as well as they have 4096 or more colors.
Other possibilities are the Enterprise 64/128, Thomson TO8, Fujitsu FM 77 AV, and Sam Coupe. I mentioned these because they all have 128 or more colors but can only display a small subset of them so changing the palette should allow color cycling.
The perspective scroll is achieved by having a colour for each 'column' of the text. The letters are then drawn by setting the palette at the start of each scanline. So for a letter T, you'd have a few scanlines with ink 1, 2, and 3 all set to green; then the remaining scanlines with ink 1 and 3 set to black, and ink 2 set to green.
Later on it varies the technique by using different shapes, by changing the colour selection per line, and so on.
You could simulate it on a VIC-20 or C64 but it's not really color cycling because you'd have to change the color RAM. The background color could be cycled but cycling one color isn't very useful.
The Atari 8-bit computers can display and cycle 4 out of 128 colors, which is quite limited.
The NES, Master System and MSX 2 all did though they could not produce images like those in the showcase. They'd need a lot more colors, so we're talking Amiga / VGA tier graphics at least.
I think of this as more something from the Atari ST era; in fact, I think I distinctly remember the waterfall effect from the NEOchrome examples: https://www.retroshowcase.gr/index.php?p=article&artid=3