Even from the start, Monster Care is very different in this game from other games in the series. You do not have your Monsters on your own farm, for a start. Instead, at the beginning, they can be housed in a Monster Barn inside Ymir, the Plant Golem.
They do not appear to need Fodder. Instead, supply them with Monster Cookies that you purchase from the Gemini General Store. You don't feed them to the Monsters. They help themselves to these treats.
The most confusing difference for me was in terms of Monster Care and Harvesting. I was unable to Brush my Monsters once I had tamed them, until I finally decided to use the option to Take the Monster with me.
As soon as we touched land on Verde Island, I was able to use the Brush on my tame Monster and it responded with an outpouring of hearts. I was able to use a Harvester as well on Monsters I had tamed only a day or two previously.
Another difference is that you can have more than one Monster with you at any given time. I had both Monsters out with me for grooming and harvesting...