J.P.
My son isn't yet 2 but still has those things that he doesn't want to go without. They alter all the time, though (sometimes it is a cat toy). I am fine with him carrying it all over the house, but it will wait for him while he is eating, and he can get it afterwards. We also don't take those things to daycare. I usually ask him to put it down on a table, and let him know that it will be there when we get back (I never move it or put it away). I also get him to say bye-bye to it. The bye-bye works too when he likes something in a store or that isn't his. I do sometimes let him take it in the car, depending what it is, but never into daycare. I ask him to place it in his carseat.
He does much better with him placing it somewhere, but as he gets into his "No" phase, the taking it away is becoming more necessary. I try to tell him that if he takes it to daycare, it could get lost. If he puts it down in a spot at home, it will there. There are still tears and upset, but not much you can do about that.