Positional. He'll take you into an endgame with a tiny nothing of an advantage and then play it so well you lose.
What is Kramnik's playing style?

That question is relative. There probably wasn't a more passive player than Tigran Petrosian, but he was also a world champion and fellow player considered him one of the hardest players to beat. I also believe he went without a loss for 2 years in the USSR championships.
Every style can be conisdered "solid" It just all depends on the style you like to play. Personally im defensive, and i wait. What i have noticed is with todays young kids who are all about playing aggressive, sacrificing, looking for tactics, is i just sit back and wait for them to make a mistake.

The attributes I hear attributed to Borvinnik have to do with his scientific approach and work ethic i.e. he wasn't as naturally gifted as say Alekhine or Capa, but he worked very hard and on all parts of his game. He's more of a universal player.

That question is relative. There probably wasn't a more passive player than Tigran Petrosian, but he was also a world champion and fellow player considered him one of the hardest players to beat. I also believe he went without a loss for 2 years in the USSR championships.
Every style can be conisdered "solid" It just all depends on the style you like to play. Personally im defensive, and i wait. What i have noticed is with todays young kids who are all about playing aggressive, sacrificing, looking for tactics, is i just sit back and wait for them to make a mistake.
Petrosian wasn't a passive player -- people confuse passive with solid, positional, strategic, defensive or quiet. Like the OP said, no GM plays passively. Find even one game for me where a GM's overall play is passive, the idea of it is silly -- especially when attributed to a former world champion.

Some people are too biased against Kramnik for some reason and they can take this to a level where they completely ignore any objective criteria. Kramnik is definitely not a "passive" player. He is a positional type of player, but he can play some nasty tactical games too.
I think his win against Anand in their WC match and his latest win against Carlsen as White in Bilbao were good examples of his positional style.There are probably lots and lots of games like these, and quite a bunch with crazy tactical complications too. These were just 2 examples from the top of my head, which were said to be played in Kramnik's trademark positional style, and against probably the best competition in the world at that moment. Note that in both of the final positions where his opponents resigned, the material was equal.

I seem to recall that both Kasparov and Kramnik have attended Botvinik's chess school and both have probably learned something about his work ethics and analytic approach. Timman once wrote that Kasparov often amazed him in post portems with the incredible deep things he had seen during games (Timman for a long time was like number 3 in the world only behind Kasparov and Karpov so he's not easily amazed). Then he went onto say that Kramink is more or less Kasparov's equal in that respect. Both are capable of calculating very deep into relevant lines that may sometimes contain very original twists.
Dvoretsky advocates classifying players according to their thinking style rather than playing style (as nowadays there hardly exists any purely positional or attacking players). So, could we say that Kramink is more analytical than intuitive player? (Other analytical players would be Kasparov, Botvinik, Rubinstein whereas for example Karpov, Capablanca, Tal or Petrosian would be more intuitive thinkers).

Agree with the above. Passive is not a word used for a GM- positional, analytical and solid are all adjectives I would use. Bareev has said that Kramnik strives for precision and because of this he sometimes neglects practical opportunities.

The fact the Kramnik plays openings like the Berlin Defence, trades queens early, and is quite happy to draw with Black probably contribute to his image as a passive player. Anand and Topalov play much more actively using pawn sacs and sharp lines to try to wrest the initiative from their opponents. Carlsen is probably somewhere in the middle, moving his pieces towards the centre in classical style, but not forcing matters unless he sees an opening.
When I was at a local club, I heard some masters calling Kramnik's playing style "passive". I disagree (No chess grandmasters are passive), but then again, I'm only 1700. So what IS Kramnik's playing style? Is he passive or solid? What other solid chess players are there? Thank you.