How is this possible?

I guess you mean pixel art? It's actually not that complicated to do, but it's complicated to make something that really looks great. I never had the patience to do anything like it.

But that being said, I'd recommend you some googeling on pixel art/tutorials. If you got photoshop, use the tool with the sharp edges, instead of the normal brush, which uses antialias. Also, looking at the screenshot, use some pastel colors, but stay within a minimal palette, which helps to sell that retro design. The last point is to scale up/rotate pixelated images inside the game. In the 4rd screenshot you clearly see that the spaceship is rotated, which wouldn't be possible with these sharp edges on an old arcade/computer. But it's clearly got the retroesque style.

I don't know if that's any help for you, but I tried ;)

curious if there were some techniques on photoshop and/or illustrator to mimic this artstyle?

You are clearly asking an art question, not a Game Design question, so I'm moving this to a more appropriate forum.

You'll probably want to use something like GraphicsGale, or even Microsoft Paint, for art like that.

That "enlarged pixel art" style is combining blocks of different sizes: The blue ship on the fourth screen has a yellow sort of "belt" that is of a different block size than the rest of it.
The energy projectiles (the ones that look like lollipops or sunflowers) have angled sides as if the pixels were rotated, something that you cannot do with pixel art.

I believe that these graphics were made with vector art or "selection based" digital painting.
The "blocky" visual is given by using a snapping grid feature to design the graphics with whatever vector art or raster art program that you might be using (Illustrator, Inkscape, Photoshop\GIMP\PShopPro with paths or selections, Xara Designer, Corel Draw etc.).

The other part is the colouring. Those samples use very bright, saturated colours - I'm sure that they were employed to show how "vivid" the screen displays of those products are rather than to be aesthetically pleasing, as that image seems to come from an advertisement.


EDIT: To prove my point, you can set up a large grid with Inkscape and have at it.

Hmm, that's really cool Kryzon, thanks for sharing.

back in the day working on gameboy, we literally just created 32 by 32 photoshop files. If we had to use a limited palette we would create a custom one

The real issue came when the agency would ask for print sized version of the sprites, i seem to remember nearest neighbour wasn't in photoshop then or didnt do it right, a real ball ache (i am talking the 1990s here)