if I analyze a game played in blitz, move by move, between two 2600s,
should I expect the moves to be of the same quality as the moves
between two 2600s in classical?
Of course, I am sure that you mean relative to the speed at which they are required to think for each time control. Faster moves are certainly going to be weaker.
As others mentioned, your rating is a measure of your strength relative to the other players in the rating pool, but even so, if you only consider a single player, and compare that person's actual strength, irrespective of rating, even then, player's strengths can vary significantly from time control to time control.
Take these two examples:
Fabiano Caruana has a reputation for being a weaker blitz and rapid player. Forgetting about specific ratings, he is number two in the world in classical, but number 12 in the world in rapid, and number 36 in the world in blitz.
There is no doubt that he has great understanding, but he also clearly, does not think as fast to be his equivalent ranking in blitz.
The second example is an older person, and I am going to draw on person experience being 58 now. I used to be a very strong blitz player (2450 on ICC when 2800 a strong GM, and only Kasparov and Shirov were over 3000...only later did ratings inflate to the 3300 range), but now, I just know that when I play blitz, my mind is just so much slower now. I can no longer compete at blitz like I once did.
My point is that your understanding of chess does not change from time control to time control, but your ability to put it into practice as quickly as blitz requires does change with various factors, but the biggest one is age for most people.
Just to be clear: You are not going to be able to play blitz-speed moves in a classical tournament as a 2600, and expect to stay 2600, unless maybe you are the next Anand, who was known for his speed of play when he was younger.