The choice of creature type is made as Metallic Mimic enters the battlefield. Players can't respond to this choice. Metallic Mimic's second ability starts applying immediately.

You must choose an existing creature type. "Artifact" and "Vehicle" aren't creature types.

Creatures of the chosen type that enter the battlefield at the same time as Metallic Mimic won't enter with an additional +1/+1 counter.

If a creature entering the battlefield won't have the chosen type until another card's effect gives it that type after it has entered the battlefield, it won't enter with a +1/+1 counter. Creatures that have the chosen type due to their own abilities, such as a second Metallic Mimic with the same creature type chosen, will enter with a +1/+1 counter.

Even though Metallic Mimic is a Shapeshifter, other Shapeshifter creatures you control won't get a +1/+1 counter unless you chose Shapeshifter as Metallic Mimic entered the battlefield. If you do, that creature must be entering the battlefield as a Shapeshifter to get a +1/+1 counter. It won't get a +1/+1 counter if it enters the battlefield as a copy of a non-Shapeshifter creature.