A bit simplified you can think of it as arrays being a special case and not ordinary classes (a bit like primitives, but not). String and all the collections are classes, hence the methods to get size, length or similar things.

I guess the reason at the time of the design was performance. If they created it today they had probably come up with something like array-backed collection classes instead.

If anyone is interested, here is a small snippet of code to illustrate the difference between the two in generated code, first the source:

public class LengthTest {
  public static void main(String[] args) {
    int[] array = {12,1,4};
    String string = "Hoo";
    System.out.println(array.length);
    System.out.println(string.length());
  }
}

Cutting a way the not so important part of the byte code, running javap -c on the class results in the following for the two last lines:

20: getstatic   #3; //Field java/lang/System.out:Ljava/io/PrintStream;
23: aload_1
24: arraylength
25: invokevirtual   #4; //Method java/io/PrintStream.println:(I)V
28: getstatic   #3; //Field java/lang/System.out:Ljava/io/PrintStream;
31: aload_2
32: invokevirtual   #5; //Method java/lang/String.length:()I
35: invokevirtual   #4; //Method java/io/PrintStream.println:(I)V

In the first case (20-25) the code just asks the JVM for the size of the array (in JNI this would have been a call to GetArrayLength()) whereas in the String case (28-35) it needs to do a method call to get the length.

In the mid 1990s, without good JITs and stuff, it would have killed performance totally to only have the java.util.Vector (or something similar) and not a language construct which didn't really behave like a class but was fast. They could of course have masked the property as a method call and handled it in the compiler but I think it would have been even more confusing to have a method on something that isn't a real class.