In the old day each new console was a very different system that was basically totally incompatible with it's predecessors, if you wanted to keep playing your old games you had to keep your old consoles around. Sometimes a game would be re-released for a newer console but this was the exception not the rule. If a game was re-released and you wanted to play it on your new console then yes you did have to re-buy it. Often such games were packaged together into compilations for the re-release.
The playstation 2 broke this trend being able to play games for the original playstation (from their original disks, no need to re-buy) with no obvious issues. The playstation 3 was supposed to continue this but the support hardware for this feature was removed in later PS3 models (some intermediate models removed some of the support hardware and replaced it with crappy emulation, later models removed PS2 support completely) to cut costs. I have no idea about the playstation 4.
Original wii models had support for gamecube games to be played off their original disks but this was removed in later models. The wii-u similarly supports wii games (but not gamecube games) off their original disks. With the wii ninitendo also introduced something called the virtual console which allows playing many of the games for older consoles (both nintendo and otherwise) but you have to re-buy them.
On the MS side the xbox 360 could play some games for the original xbox. You could use the original disks (didn't have to re-buy) but you did have to have a hard drive and internet connection.
I belive sony and MS are doing something similar to the virtual console but I haven't investigated the details. I'm also not sure what is going on with the PS4 and XBONE as I lost interest in consoles before they came out.
And yes many people do build shelving around their TV to support their collection of games consoles.