No, 5e CR does not map the same way as 3e CR.
I am looking at a pregenerated 1st level level fighter, who will have AC 18, HP 12, Attack bonus +5, damage ~8. That appears to have defences equivalent to a CR 1/2 NPC/monster, and attacks equivalent to a CR 1 NPC/monster, which I think means it is midway between CR 1/2 and CR 1. Let's call it CR 1 because of fighter bells and whistles.
Now, looking at a pregenerated 3rd level fighter, with AC 18, HP 28, Attack +5, damage ~8. It has CR 1/2 defences and CR 1 attacks, which still means it is still CR 1 (5e is not very granular).
Same fighter at 6th level: AC 19, HP 52, attack +8, damage ~10. It has CR 3 defences and CR 3 attacks, and appears to be CR 3.
Same fighter at 10th level: AC 20, HP 94, attack +9, damage ~10. CR 5 defences and CR 4 attacks, so it is midway between CR 4 and 5, and probably counts as CR 5.
Compare this to what happens if you try the same calculations using encounter tables. Correct me if I am wrong, looking at the 3.5e tables, if a group of 4 1st level fighters battles a group of 4 CR 1 monsters, that would have an Encounter Level of 4 and be at the top range of "very difficult" encounters. That "very difficult" rating appears to be the case whenever PC level = CR and you have the same number of creatures on each side.
If I am reading the 5e encounter building rules correctly, an encounter that was "hard" bordering on "deadly" for 4 1st level PCs would involve 4 CR 1/4 creatures - significantly less than the CR of 1 we estimated above. But after that it appears to track more closely. For 4 3rd level PCs the equivalent encounter would use CR 1 NPCs/creatures (or a mix of CR 1/2 and CR 1), which is similar to what we calculated above. For 6th level PCs, the equivalent encounter would use CR 3 NPCs/monsters, and for four 10th level PCs it would be about right to use two CR 4 and two CR 5 NPCs/monsters, both of which track with what we calculated above.
So what I am getting is PC level 1 = CR 1/4; level 3 = <CR 1; level 6 = CR 3 and level 10 = CR4 to CR5. But I don't have enough play experience to know if it works out that way in practice. And the whole thing is incredibly non-granular, with huge jumps in power level between one CR and the next.
I mean, I have this theoretical map of equivalencies based on what the edition says about itself in its own encounter tables. It looks to me like CR 1/4 is a bit below level 1 (say the equivalent of 3e CR 1/2)
CR 1/2 = level 2
CR 1 = level 3
CR 2 = level 4
CR 3 = level 5
CR 4 = level 8
CR 5 = level 10
CR 6 = level 11
CR 7 = level 12
CR 8 = level 13
CR 9 = level 15
CR 10 = level 16
CR 11 = level 18
CR 12 = level 19
CR 13 = level 20.
But I have no way of knowing if that works out in practice.