Does a copy of your commander count as your "Commander"?
No, only your own original commander counts.
Generally, if the Lieutenant ability refers to your Commander, then the opponent's Commander would not count, since it's not yours, even if it has the same name. Also, a clone of your Commander would not count, because the Commander-ness of a card cannot be copied.

903.3. Each deck has a legendary creature card designated as its commander. This designation is not a characteristic of the object represented by the card; rather, it is an attribute of the card itself. The card retains this designation even when it changes zones.
Example: A commander that’s been turned face down (due to Ixidron’s effect, for example) is still a commander. A commander that’s copying another card (due to Cytoshape’s effect, for example) is still a commander. A permanent that’s copying a commander (such as a Body Double, for example, copying a commander in a player’s graveyard) is not a commander.