Heat oven to 375ºF.
Heat the olive oil in a medium saute pan over medium heat until shimmering. Add the onions and salt, and sweat until the onions turn translucent, 1 to 2 minutes. Do not brown. Add the garlic and continue to cook for 1 more minute. Transfer the mixture to a medium mixing bowl and set aside to cool.
Clean the squid thoroughly and remove the heads from the tentacles and discard the heads. Turn the tubes inside out (since the outside skin tends to curl outward, it'll hold the stuffing better inside out), put into a bowl and set in the refrigerator until ready to use. Place the tentacles and the shrimp into the bowl of a food processor and pulse until there are no large pieces visible, 6 to 8 pulses. Do not process until smooth. Transfer to the mixing bowl along with the onions and garlic. Add the bread crumbs, tomato, lemon zest, ginger, parsley, and pepper. Stir to combine well.
Place the mixture into a resealable freezer bag and snip 1 corner. You may also use a pastry bag with a tip just small enough to fit into the end of the tubes. Pipe the stuffing into the tubes, dividing the mixture evenly between them. Do not overstuff.
Place the tubes into an 8-by-11-inch glass baking dish and cover with the tomato sauce. Cover tightly with aluminum foil and bake for 30 minutes. Serve immediately.