Shenmue was insanely cool when it came out in 1999. To consider Ocarina of Time came out the year before, open world games were a brand new thing. Shenmue paired that with an epic action life-sim. It was just so unexpected; so many minigame-like mechanics for doing things around the world. It was incredibly immersive and ahead of its time.
I don’t think it has as big an impact on players as it had on developers.
The Yakuza series sticks out as greatly inspired by Shenmue / spiritual successor, as with Persona series. Also games like Lake, Fahrenheit / The Indigo Prophecy, GTA/RedDead, FFVII Remake, all owe a little to Shenmue.
Dreamcast was the best console of the sixth generation. If Sega hadn’t spun up MegaCD, 32x, and Saturn in such a short time, I think developers would have embraced the Dreamcast more rather than be weary Sega would abandon it for another system a couple of years later.