What's wrong with usage of word "different" in this particular way/style?

There's nothing wrong with using different. Microsoft has simply put certain rules into its software. In addition to catching typos and obvious grammar mistakes, it tries to "nudge" you into using certain language.

In this case, it likely sees "types" (or "types of animals") and is trying to get you to use more formal language. Using "different types of animals" in a scientific paper, for instance, is, I suspect, less common than one of the options it's proposing. (Although, if it does the same thing when that's the only word you type, then it has nothing to do with context at all.)

What I find more strange than the simple fact that it's doing this at all, is that it thinks several is better than different. I can understand the other two, but not that one.

In the end, there's nothing wrong with different—it's just more informal. I wouldn't trust Microsoft's style decisions at more than a basic level. See what it suggests and then accept or ignore it as is appropriate.