Here's a picture of the head-unit, maybe?[0]

or an earlier one? [1]

[0]: https://i0.wp.com/www.curbsideclassic.com/wp-content/uploads...

[1]: https://www.thetruthaboutcars.com/wp-content/uploads/2014/10...

DSPs were around, but there's no way that car stereo was digitizing the signal, delaying it, then converting it back to analog.

I get it was an impressive experience, but it's essentially certain it's what the other poster said: just boosting the out of phase content between the channels. This was a very in vogue effect at the time. I remember listening to the top 40 on the radio one time and Madonna had some new song where they were hyping it as surround sound and turning it into a whole event. In any case this effect can be more effective than you might assume. After all, the first consumer version of Dolby Surround was just this out of phase content run through a bandpass filter and sent to surround speakers.

I knew a family friend with the 90s version of the Impulse. Neat quirky car from what I remember. As a kid I definitely thought it was very cool.

Sounds like q-sound. There was a Rodger Waters album mastered with it (Amused to Death) that has a dog barking way outside the normal sound stage.

Do you mean that they simply amplify the difference between the two signals?

Yup, that's the basic idea. It's a very simple circuit, which was the appeal before the digital everything era we live in now.