Anyone here familiar with the Japanese game known as kokkuri which is similar to the Western table tipping? Does the term "kokkuri-san" refers to a specific race of yokai or it could be any race of yokai summoned by the game?

Yes, I know this 'game' It isn't a game at all. A Kokkuri board is like a Ouija board... They can be used to summon good OR bad spirits. I would highly recommend not messing around with it at all.

Yes it's "like" the Ouija board, but at the same time - no, it's nothing like it. It's not ghosts (apart from one version) firstly, and the spirit that comes through either all have a roll (three-being version) or is a neutral (fox version) that COULD have the possiblity of having a very negative effect if you don't do the right things.

I'm familiar with Kokkuri-san - is what I should say. What it is depends on the version - some say Kokkuri-san is a fox (its name is Kokkuri), others say it's three beings of a high level such as messengers of the gods (the fox (ko), dog (ku), and moneky/tanuki (ri) - which is why you use a three-legged table), and others say it's even the spirit of a child.
It's pretty dangerous, and much like the Ouija, but not to be mixed up with. In the three being version, the fox is the most dangerous (apparently), the dog is the most docile and most willing to help/protect you, and the monkey/tanuki is the trickster, and you don't know which of the three (or even if all three) has come through. The fox-only version, you should always have something like Inari-zushi (aburaage foods) there, so that you won't be cursed, etc. from using the Kokkuri-san. It's particularly seen as dangerous 'cause you're coming to know things you shouldn't, or in ways you shouldn't
Also, you should never do it alone, or you're more vulnerable to the bad things that could happen. The things that could happen range from being curse, becoming sick (possessed), disappear, or even driven made/to death. Those that play alone sometimes kill themselves or try to kill others. There's rules about what you should do with it afterwards, too - such as, in the three days after you use it, you should purify it in salt water, burn it, or cut it up into 42 pieces, etc. Getting rid of the paper you used doesn't protect you from the curse, however, and there are ways of getting rid of it... but best just not using it unless you SERIOUSLY understand Kokkuri-san (and how it's not Ouija), and are aware of what you should do and the dangers in using it...
I had more information, but my browser crashed on my twice while writing a comment, so I'll leave it there and add anything if I remember (or if you have questions).