While farming in a Survival Minecraft world, I had accidentally switched my stone axe to my offhand by clicking F. I didn't want the axe there, so I pressed F to switch my axe back to my offhand. In doing so, however, I noticed that it took longer for the axe to reappear on the screen than it took for it to reappear during the accidental switch of the axe to the offhand. I was then intrigued, so I tried the same process with a block of dirt. I was surprised to find that the block of dirt, rather than the axe, took the same amount of time to switch to the offhand as it took to switch back to the main hand. I tried with my other stone tools, and found that they all behaved similarly to the axe.
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And by the way, replying to your message, you said, "Different attack speed, it's a lot faster." in response to my "Then why did it work with my stone hoe?" By the "Then why did it work with my stone hoe?" I meant "Why did the stone hoe behave similarly to the axe?"
Axes and all other tools/weapons switch to the off-hand quicker because the player cannot perform attacks with them in the off-hand, that is why they are slower (or at least the axe is) when switching to the main-hand. Switching any item to the off hand takes the same amount of time.
Axes have an attack speed, the attack speed charge is indicated by the item moving up. It only applies in the main hand as you cannot melee attack from the offhand, and is not exclusive to swapping hands (deselect and reselect, attack once, whatever).
Not a bug.