Well, generally FM radio stations are highly EQ'd, especially on the bass, which may contribute alot toward that "wide fat" sound you refer to. As to the vinyl, I don't have an explanation for that.
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Perhaps the dynamic range is being compressed when you record to tape? This would make the softer sounds more audible, and if those previously softer sounds are more channel specific, you could experience a greater stereo ("wider"?) effect. And perhaps "fatter" comes from any equalization the signal goes through in the tape deck.
If it turns out to be the compression that you like, you might try one of the little dbx compressor/expander boxes to modify the dynamic range when playing vinyl. And as long as I'm disregarding signal quality, perhaps an equalizer could mimic the signal change of the tape deck?
The effect you describe is called tape compression. Many people do tend to prefer that tape effect over digital...that's why many producers and engineers still prefer to work reel to reel recorders rather than digital, and many more use some combination of the two.
As far as the FM goes...the signal could be compressed, it could be the CD player or turntable they used, etc.