You also had GoldSrc, a modified Quake Engine for Half-Life 1.
Many companies created their own and was not available outside the business. Dark Engine used in the Thief:Dark Project games is an example. Think of companies like Nintendo having their own engine which powered games like Mario 64 and, I believe, Ocarina of Time was a modified version of that.
The "off the shelf" engines available at that time (like Unreal or Quake engine) might have been decent for certain type of games.
Today, you could say many games could be done with Unity, Godot, or Unreal. There are still companies today using their own.
Source itself being built from Quake 1…
They had already done licensing deals for the DOOM engine at that point, including the greatest game of all time "Chex Quest."
No, this isn't true. Almost all games were bespoke back in that era - the machines simply weren't powerful enough to allow for competitive, flexible game engines. An individual development house might have a code library or base that they'd iterate on, but there was little sharing between different companies (much less reuse by the thousands).
By the 16-bit era sound engines did tend to be widely reused, though (e.g. GEMS).