That's a common naive argument but
anybody with ears should know it's not true, except the "kids" who haven't heard vinyl.
Some people prefer the sound of vinyl and that's OK, but it's technically inferior and it has less usable resolution than CD (or MP3).
As kemmler3D said, the resolution of analog is limited by
noise.*
It's like saying (analog) VHS tapes have more resolution than (digital) Blu-Ray or that an analog tape measure has more resolution than my digital calipers.
The differences ARE audible!!! You can always hear noise between tracks and during fade-outs (especially with headphones or when the volume is turned-up). And often there is "snap", "crackle", and "pop". Most records have at least a few clicks pops and sometimes some nasty-loud clicks pops. I grew-up with vinyl and those clicks & pops always annoyed me, especially when it was my record... I'd know when the click was coming and I'd be waiting for it rather than enjoying the music. (Most people didn't seem bothered by it.) When I got my 1st CD player I was amazed by the dead-silent background!
Records (and phono cartridges) also have frequency response irregularities, which aren't always a BIG problem because you can use tone controls or EQ, but digital is "dead flat" over the audio range. And you occasionally get tracking distortion which doesn't happen with CDs (or other digital).
Having grown-up with analog,
digital is wonderful!!!
* Interestingly, if you listen to an 8-bit file, the lack of resolution is heard as quantization noise, even thought it's different from analog noise and unlike analog noise it goes-away completely with silence.