After digging through TONStons of MS-DOS game screenshots, I found it!
It's Engineering Jones and the Time Thieves of DSPea.
The story behind this game's development is interesting. It's actually a 1991 advertisement for Harris Semiconductor DSP chips. The game was apparently mailed to customers, and those who completed it had the opportunity to enter a drawing for a Sony CCD-TR4 Handycam CamCorder. It was distributed on a floppy disk (not sure what size), and Harris encouraged customers to share it with their colleagues. This may by how my friend got it - someone may have given him a copy. Very interesting find overall. I am impressed with the quality even though it is an advertisement.
Here is the Mobygames description:
In Engineering Jones and the Time Thieves of DSPea the Time Thieves, plunderers of time-changing technology throughout history, have hit the city of DSPea and stolen their cutting edge image processing technology. Engineering Jones, Senior Time Warden of the sector, along with his robot buddy CirQuet, is charged with setting history back on track and stopping the Time Thieves.
Engineering Jones is a first-person maze game meant to advertise Harris Semiconductor products. There are also puzzles or mini-games with a different type of gameplay. People who managed to beat the game could participate in a contest to win a Sony Camcorder.
Here is a video of a full playthrough:
And here are relevant screenshots from my memories (courtesy of Mobygames).
The futuristic city:
The interior of the futuristic city:
The Time Thief aliens stealing a DSP chip (not a floppy disk) in the game's introduction (not exactly Pac-ManPac-Man ghosts, but I did remember a wraith-like design at the bottom):
And the top-down section with a crumbling bridge made of octagon-shaped tiles: