A player may swing the Engineer's melee weapon much faster than expected, if they use this exploit. From what has been observed, it looks like the "swing timer" is paused by hauling any building, and then it resumes as soon as the Engineer in question stops hauling the building in hand. Given that most melee weapons take time to swing before registering a hit in the world, "swing timer" is the term I use in this issue.
This paused swing timer may be combined with holding primary attack, which leads to two swings; the paused & resumed primary attack, and the normal, expected, primary attack. The normal expected primary attack ignores the resumed swing timer, which leads to two attacks stacking up on each other.

Exploit in practice (0:29): https://www.youtube.com/watch?v=m-ZX6fZ6_FA&t=29s
Demonstration video: https://www.youtube.com/watch?v=RHJOVRdeGcE

The first video shows that this exploit may be used in Mann vs. Machine as well, and combines well with the Engineer's ability to deploy hauled buildings in an operable state instantly. Experienced Engineers may "unwrangle" Sentry Guns, skipping their inoperable state debuff after using the Wrangler, by simply picking it up and place it back down. If they were to do this trick while performing the exploit, they hit their Sentry Gun twice in under a second, regardless of attack speed upgrade for the Engineer's melee weapon.

The second video demonstrates how the exploit may give unfair advantage towards enemy players. During this demonstration, the hauled buildings are destroyed with the help of the "destroy" command. Later on in the video, the Gunslinger is brought forth to demonstrate how the exploit may be lethal to most enemies, given the circumstances in the battlefield are correct. One can say that the three-hit combo challenge, for a guaranteed critical swing, for the Gunslinger may be trivialised by this exploit.

Steps for reproducing the issue:

1. As the Engineer, with the melee weapon in hand, hold primary fire and secondary fire, while in the pickup range for your own building (any building).
2. If the melee swing animation plays, but the building is being picked up, the reproduction of this exploit is done correctly.
3. Either place or destroy the building that is being hauled.
4. After placing/destroying the hauled building, the "paused swing" will resume. Press primary fire button to add another swing, therefore stacking two melee attacks.

This is all I know about this issue. The aforementioned "swing timer" is a speculation.