Well, if Pokemon were real, I'm pretty sure our lives will be radically different from what we are used to. So many of aspects of our lives will be different in obvious and very subtle ways. I'll start with the most obvious: [This is the sparknotes version of what I have in mind]
Pokemon Journey: While I'm sure that the age to raise Pokemon would be higher than 10, I'm pretty sure there will be plenty of high schoolers will go out to Pokemon journeys and get Pokemon badges. As usual, the starters will be always hard to choose for each individual but overall, there would be a good number not being in high school, at least right away.
Given the number of Pokemon trainers that will be going on journeys, there would be lots of gyms instead of the usual 8 gyms per generation. The trainers would be required to defeat a certain amount of gym leaders in the region to qualify for the Pokemon league. I'm sure as time goes on, qualifying for the Pokemon league would be more and more difficult. Of course, I'm pretty sure most of the trainers would just go on a journey for the sake of going on a journey.
Non-trainers: For the vast majority of the population, they are or will be keeping Pokemon as pets. However, they will also be using Pokemon to run their day to day lives. For example, I'm pretty sure that a majority of fire Pokemon would be great to keep the house warm, electric Pokemon for electricity, water Pokemon for main source of water and so forth. It'll get to the point that having a Pokemon will be required.
Politics and Government: At least in relativity civilized parts of the world, there would be public policies that would deal with the interaction of Pokemon and humans. I would have to say that laws protecting Pokemon would be more protective than what we have for animals right now. Of course, the issue if Pokemon have human rights would need to be addressed, especially with the large diversity of Pokemon and abilities that they have.
Resources: This is where one of the most interesting aspects of society that will be radically different and is hard to pinpoint what it would be like. However, we do know that resources is [virtually] unlimited now that we have Pokemon and that many of the resources problems will not be as big as a problem as they are right now for us without Pokemon. I can imagine a world without too many big problems with high amounts of water and electricity.
Science and Technology: I'm pretty sure we would be much more advanced than we are right now. I really don't know how I can put this in a concise manners without getting into too much details but all I can say is that the field of biology, especially at the molecular and ecological level would be far more interesting [so that more people would be engaged] and more advanced that it is right now. I can imagine that Pokemon helping humans by just assisting them in research and being subjects of both ethical and unethical research.
Overall, if Pokemon were real, we would be living in a much different world. Certainly the world will be more different than just school age kids going on Pokemon journeys. We have solved most of the world's problems with Pokemon, but at the same time, we would have to deal with other problems that are more than what just the game presents.
Sorry if this doesn't make too much sense and would be more than happy to clarify of aspects that I posted that weren't cleared. It's just hard to put "how the world be different with Pokemon" all in one posts and in a concise manner.