For me it was Fallout 3. I played exclusively on console growing up but after seeing a video on YouTube about custom player homes I felt like I needed to give it a try.
Fallout 3 did allow the player to theme their house but if you really wanted to fully decorate it to have a lived in/cluttered look you had to drop items out of your inventory and make them kind of awkwardly hover in front of you. If you wanted to rotate them you'd have to carefully push it against something like a shelf until it was oriented how you liked it and hope you don't knock anything else over while placing it. Having things kind of welded in position or pre-placed was a big time saver.
The photo attached is from the Underground Hideout mod. It was one of the first mods I installed.
The earliest thing in my experience is probably modding GTA 3 on PC as a kid. I remember discovering some of the game configs had editable plaintext form and I used that to mess with NPCs, like making them jump really high, infight each other, or giving cops a hoe AI so that I could pick them into my car and make them do the thing. Later I discovered their textures were also easily editable and made some really ugly edits in paint. I was also into weekly/monthly gaming magazines, some of them came with CDs and they started putting actual mods there so I started installing those. I think it was still dial-up times when this started, getting mods from internet came later. I even made a website with some GTA 3 mods/skins back in the day. It was ugly as hell, and I was like in 5th grade in school.