The adoption limit would be the max of how many you can get from the
'Adopt' page. For example, this month the adoption limit for the dog litter is 6, if you try to adopt more than that it won't let you and will say you have reach the limit. You are allowed to trade for any many extras as you want outside of that though. There isn't any kind of limit on how many pets you can have or how many of a certain pet, the adoption limit is just a limit on how many you can get for free without having to trade with anyone.
The reason you can adopt so many pets is because they do not all grow into the same adult. Monthly pets always have many different outcomes they can grow into. So if you only adopt one of each pet, you'd miss out on a lot of outcomes. The adoption limit usually reflects how many outcomes there will be, for example, if you can adopt 6 that usually means there will be 3-4 different outcomes from that litter, so adopting the max gives you the best chance of getting all the outcomes. You many still miss some of the outcomes, but then you'll have extras that are duplicates which you can trade for ones you missed.
If you are interested, you can see the current growths for May over on the outcomes thread;
Forum/viewtopic.php?f=20&t=4959766They will also appear in the archives eventually, but the archives is only updated once pets reach their adult stage so it doesn't show everything until the end of the month and doesn't show any that are still growing.