When a 3 might unit attacks a battlefield that has 3 enemy units, can it distribute the 3 might onto 1 each?

No, you cannot distribute the 3 Might onto three 1-Might units in that way.

According to Rule 460.2.c.3, "Units must have lethal damage assigned to them in full before damage is assigned to a different Unit."

Lethal damage is defined as any non-zero amount of damage equal to or greater than the unit's current Might (Rule 460.2.c.2). Because your attacking unit has 3 Might, and the defending units have 1 Might each:

1. You must first assign 1 damage to the first defending unit to satisfy its lethal damage requirement.
2. You then have 2 Might remaining to distribute. You must assign that remaining damage to the next unit(s) following the same lethal damage rule.
3. You can kill at most three 1-Might units if you have a total of 3 Might, because you assign 1 damage to the first (1/1 lethal), 1 to the second (1/1 lethal), and 1 to the third (1/1 lethal).

However, you cannot choose to assign less than the unit's Might to a unit to "spread it out" if that amount does not reach the lethal threshold. You must kill a unit completely before moving your remaining damage to the next target.

As confirmed by RiftJudge FAQ #1348, Imperial Decree and similar effects do not change the requirement to assign lethal damage to a unit in full before moving to the next one.