Answer
Yes, when traveling by ferry between Argentina and Uruguay, you need to account for two border crossings: this is an international route, not an ordinary walk to the pier. Border control begins before boarding the ferry.
That is why the ferry cannot be treated as domestic transportation. You need a valid travel document, permission to cross the borders of both countries, and enough time for formalities, queues, luggage, and boarding, especially on weekends and during high season.
For a short trip to Colonia, this is especially important: the old town is close to the terminal, but a delay at border control can eat up part of the day. It is more convenient to choose ferry departures so that, after arrival, there is time for a leisurely walk, museums, and the waterfront, rather than just the return check-in.