It’s a slightly fiddly question of terminology.
First of all, a “lap record” is the fastest lap set by a driver during a race.
The records that you’re looking at from the 2004-06 era are not being beaten in races, they’ve being beaten, as you say, in qualifying or other sessions, so you’re not comparing apples with apples there.
It’s not an uncommon area of confusion. In an attempt to clear it up, in the circuit information pages on this site I refer to both the lap record and the fastest lap set at the track during an F1 race weekend. Here’s the information for Monza, for example:
Autodromo Nazionale Monza, Italy – circuit information
As you can see for this track, both the fastest lap set during the race and in another session were set during 2004.
This also reflects the example you picked out for Monaco:
Monte-Carlo – circuit information
There are several of reasons why lap records generally aren’t falling at the moment: the cars have smaller engines, aerodynamics are more restricted and there’s no tyre war. Plus, the banning of in-race refuelling means race lap times tend to be slower. More on that here:
The end of the pursuit of speed
I hope my reply made that less and not more confusing!