If I understand the concept of "partial types" correctly (from the C# example), in Perl 6 this simply looks like that:

class Foo {
   # code here
}

...
class Foo is also {
   # more code here
}
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And they are usually called "open classes".

Why we want/need them?

Suppose you are not happy with the Perl 6 prelude, and want more methods in "basic" types:

class Int is also {
    method frob {
        say "frobbed {self}";
    }
}
4.frob;
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Isn’t Multiple Inheritance (evil?!) some kind of partial typing?

Multi Inheritance isn't evil (it's only evil if your tools aren't good enough). And it's different from partial types that it doesn't extend existing classes.

How does this compare to multiple-type objects?

What's a multiple-type object?