My stepsons were (and still are) very different from each other. Granted, I didn't meet them until they were 5 and 6, but their dad will tell you that they were different from Day 1. Son #1 - laid-back, easy-going, Mr. Go With The Flow, hardly ever gets upset about anything. Son #2 - total opposite, square peg in a round hole, disagrees with everything, always wants something different from what everyone else wants. My husband joked that Son #1 didn't cry for the first 3 months of his life while Son #2 came out screaming and crying and still hasn't stopped - and they are now 17 and 18!
Our daughter is 4 and she is the very definition of spirited and strong-willed. She is easy-going and agreeable when she wants to be, is an absolute sweetie-pie (her preschool teacher told me she is the happiest and most well-adjusted child she's ever seen!), but when she gets it in her head that she wants something, or wants something a certain way, there's no convincing her otherwise. I have found that in those times, it helps to give her limited choices, but also let her know absolutely what is and what is not okay. I have to be very, VERY firm and consistent with her because wavering from the rules once will mean she will want it that way every time and she will try to wear me down with all she has to get me to change my mind. For example, the other day she wanted to wear a dress to school that was more appropriate for warm spring/summer weather, and right now it is February in Michigan, so you can imagine it is not very warm at all. I told her she could wear the dress, but she had to layer underneath with a long-sleeved shirt and some leggings or tights. No, she just wanted the dress, and she argued that she would be fine, and she had "a better plan." It didn't matter that it was 35 degrees out, she didn't care. So I simply told her, you wear it with the layers, or you are not wearing it at all. And if she didn't make up her mind to wear something appropriate, she was not going to school, period. It took her 10 minutes of whining and pleading but I just remained calm and stood firm - these are your choices, take it or leave it. Eventually she relented and agreed to the extras. There are also many times she gives me a hard time about what is for dinner, because she would rather just eat PBJ or pizza or a hot dog. Nope, this is what is for dinner. If you don't want to eat it, you don't have to, but I am not fixing something else. Sometimes it is an hour before she decides to eat but eventually hunger wins, and she realizes I am not changing my mind. If she freaks out because we are leaving somewhere and she doesn't want to go yet, we leave anyway and she is told we are not coming back anytime soon because she is not cooperating with me right now. Then when she asks me at some point can we go to XYZ, I can tell her, no, because you weren't good about leaving last time, and I am not going to do extra fun things with you if you are not going to be good for Mommy. I've found that, now that she is 4, it helps a little to let her talk and state her feelings and let her feel like she was been heard, but I still have to let her know that it's okay she feels how she feels, but it's not going to chance what I do or what is going to happen.
You are not going to be able to treat them the same because they just are NOT the same! What works with one will not necessarily work with the other and they need to be treated and honored as individuals. With him being 2, some of what you described really is just him being 2 - like thinking that Mommy can somehow magically fix the popsicle. My daughter could be like that too at that age - if she wanted blueberries for a snack but I didn't have any, she still thought somehow I could just make them appear. It didn't matter how many times I told her I had to get more from the store, she was still convinced they were in the house somewhere. Now that she is 4, she gets it. Sometimes it just takes time.
If he is really tantruming, the best thing you can probably do is leave him alone (as long as he is not someplace where he can hurt himself) and let it blow itself out. You can't fix everything and you can't always make them happy - sometimes the less you come to their rescue, the more resourceful and independent they become. They learn to be happy on their own, which in the end, is what you really want. It is going to be rough few years but the more you accept your son for who he is, and appreciate the traits that he has, the easier it will be. A couple of books that really helped me in the discipline department were "1-2-3 Magic!" by Thomas Phelan and "Disciplining Your Strong-Willed Child" by Robert MacKenzie.
As hard as it is now to deal with it, that strong-willed personality will most likely pay off for your son in a big way down the road! Like my daughter, he will be the kid who doesn't just take no for an answer, who does not give up so easily, and is willing to stand up for themselves and what they believe in. The key is trying to get it under control now just enough so they are manageable while they are still living under your roof, and channeling that drive into something positive. Hang in there and good luck!