There are ways you can change this. 3E has clearly monsters that are a lot more unique. Fighting a Beholder or fighting a Giant are two very different experiences. They have unique innate abilities, and players will have to account for them differently, entirely based on the monster, not on class levels.
The reasons some monsters are unique and some are not have a lot to do with the exceptions they have built in. A Hydra can make multiple attacks even if moving, and its head regrows. A Beholder fires multiple eye rays and has a magical cone. You can multiclass as much as you want, you will not get this abilities. Monsters like these are examples of "exception based design". Normally, no one has access to antimagic cone. Beholders do. No one can just regrow his heads, and get even more then he used to. Hydras do.
Dealing with a Hydra requires to find a way to deal with Regeneration (and utilize the option to cut down heads). Dealing with a Beholder requires to deal with antimagic-cones, and multiple magical attacks. You need different, unique tactics to deal with these threats.