Maybe I wouldn’t call them Quake rips though; However, all the Unreals certainly weren’t innovative in any meaningful way, gameplay wise.
The engine business exploded for Epic around Unreal Engine 3, I believe. Before that: they really wanted to sell those games. See the career of aforementioned CliffyB – he wasn’t Epic’s John Carmack, he was their John Romero.
As for Id Software, they did licensed their technology since early days (Half-Life, Call of Duty, Medal of Honor, few others), but indeed Unreal surpassed them with tooling and support.
Id tried to recover their position, around Doom 3 they renamed their „Quake engine” to Id Tech, and was working on the user-friendliness of it’s offering. I think they abandoned that idea by themselves, nonetheless, soon after they were bought by ZeniMax and that was it: no more licenses for companies not related to Bethesda. They don’t even bother to allow their current games to be modded.
Meanwhile Epic’s engine business was flourishing, yet the games were flopping. Notice that only after Fortnite’s success they decided to open their Steam rival.
Had they done those the other way around, it probably wouldn't have worked- Fortnite is what sent Epic Games from the technosphere to Wal-Mart in terms of societal recognition.
No meaningful way, gameplay wise? UT had way more game modes than Quake3 did... two-team TF2-style objective with attackers and defenders, that football mode, betrayal mode, four-team domination, or the various vehicle modes... I had a lot of fun in those modes which they seemed to enjoy experimenting in
Watch one of the Fortnite world cup matches going on right now [1] and tell me that you're watching a pubg rip :)