And that sound is exactly what it seems modern popular mastering - at the request of the client, or the mastering engineer's personal technique or style - is after. While listening to NY City's Z-100 CHR FM station, Adele's 'ROLLING IN THE DEEP' came on. I quickly got up, grabbed my '21' CD, and put it in the carousel. Sure enough, that same song playing from my CD sounded *almost* as loud as it did when switching between it playing on the radio station and on my CD deck. This experience recalls an anecdotal tale I recall from the early 1980s, when Carly Simon referenced that same Z-100 station in a conversation with her engineer regarding how she wanted her then current album project to sound. Not a good role model I think - a major metro Contemporary Hits Radio outlet, that is!
My compliments on attempting the comparison!
I'm not completely discounting your effort, but please recognize the flaw here. An FM station must limit "peaks of frequent recurrence" to a level defined as 100% modulation. All stations are dealing with the same rules in the US. The loudness war is a fight for the highest apparent loudness without exceeding legal maximum peak level. Competition in the market drives every station of a given music genre to attempt loudness equal or greater to it's neighbor up or down the dial. That maximum peak level is pretty hard and firm, but average or RMS loudness levels are not regulated at all.
On a CD, the maximum is also there, 0dBFS. But there's rarely an instant comparison between CDs, like pushing the preset tuning buttons on a radio. Still, it is the same war.
The problem is, we don't know what specific level 0dBFS is in any system (we just turn the volume control) and we don't know what specific level 100% FM modulation is. They come from two different devices, and are almost certainly not the same. So a direct loudness comparison is not actually possible without a LOT of setup effort. Confounding the comparison further, the station is applying a rather huge amount of its own processing to a tune that has already had loudness processing applied. Part of the station's processing chain involves dynamic equalization, which is a large part of what makes up the "signature" of the station. It's applied in multiple bands, 2 to as many as 30, but typically 5 or 6. With all of that, plus broadband compression, high frequency limiting to protect from overmodulation, peak limiting, and yes, clipping, you'll have only two distantly related results.
IF they were similar, that actually says something for the station. But it could also be accidental. More likely, the FM station will sound duller by comparison, and with less separation, unless you're listening to HD radio. Duller comes from trying to control the HF boost of pre-emphasis, which is 17.5dB @ 15kHz re: 400Hz. You don't have that on CD or HD radio. Hence the duller result on FM.
Oh, and FM has it's own kind of anti-aliasing filter between 15 and 17kHz, keeping audio away from the 19kHz pilot tone. Those filters have always been a problem all their own.