I think it sounds really cool.
I've tried to research on this and found what's called the 'tack piano', but that doesn't sound the same. If someone can explain to me about this, that'd be appreciated.
I think the effect you are hearing is "Wow and Flutter" which is caused by a mechanical imperfection in the rotation speed of old turntables and tape decks.
Wow describes the sound speeding up and slowing down in a regular way usually caused by off center or warped records, but it can also come from worn out equipment.
Flutter describes the warbling sound quality which is more random in nature and can come from worn motor bearings or old tapes that have become stretched unevenly over many years of use.
All mechanical audio systems suffer from these problems to a greater or lesser extent.
It is possible to simulate the effect electronically - a quick google found this VST plug-in version that looks pretty interesting http://www.vst4free.com/free_vst.php?id=602 - but I'm sure a more detailed search would turn up other options.
It's weird that audio engineers of the past put so much effort into minimizing these effects only for modern day programmers to want to reintroduce it artificially. But there is definitely a warm nostalgic quality to those old recordings.
answers from Todd Wilcox and Noel Waters are essentially on the mark: you're hearing a number of different artifacts, some of them very old, some of them very new!
The original recording has some flutter. It's likely taken from the optical audio track of a 16mm film reel, and every time the sprocket holes engage with the gears, there's a little jolt. Also, you'll notice that the mics that are up are ribbons, which likely resulted in a "not as bright as we like it today" recording. That's important for our next issue: Remastering.
Someone did some "forensics" restoration to this bit, brightening it quite a lot, which brought out the brittle frequencies of the piano, which itself sounds like it wasn't precisely in tune at the session.
After that, it was mp3 encoded, which wreaks havoc on enhanced high-end of old recordings, so you're hearing mp3 artifacts as well.
I'm pretty sure what you're hearing on this recording is a combination of the piano strings being slightly detuned and also the age of the recording adding a warble to the overall sound. It's possible that the piano was tuned perfectly and all that you're hearing is the recording quality being bad, but if you want that kind of sound, you get it by detuning the strings slightly (or asking your piano tuner to do that for you).