Yes, fermentation does produce CO2, but it is not always easy to capture that CO2. In the traditional method champenoise process used to produce champagne and sparkling wines, the wine is carbonated by adding a dosage of sugar and wine to trigger a secondary fermentation that provides the tiny CO2 bubbles for which these wines are famous. In the industrial charmat method used for cheap sparking wines, the still wine is carbonated using bottled CO2. The result is larger bubbles.
I once worked in a paper mill that used purified calcium carbonate as a filler in the paper. There was a plant adjacent to the paper mill that calcined limestone to produce calcium oxide lime that was then mixed into water. We then routed exhaust gases from one of the boilers so it could bubble through the mixture. The CO2 from the exhaust gases reacted with the lime and pure calcium carbonate of precipitated out. Since CO2 was released when the limestone was calcined in a kiln, there was no net reduction in CO2 emissions.