Status display module

Your code has somewhat inconsistent styling which can inhibit readability and prevent the code from looking crisp and professional. For example, some of your function definitions have braces on the next line others don't. Some of your function definitions and calls put spaces between parenthesis var signal = function ( status ){, and some don't var inject = function (functionToReplace, injectedFunction).

Also, I don't believe that the last line init( constructorOptions ); is executing because you return from the constructor in both the if and else blocks right above it.

That said, using closures to keep private functions private is a great idea.

As for the modularity, while returning a separate object for for testing is a truly fascinating idea, my gut instinct goes against it. I find it somewhat troublesome because it violates cohesion with your constructor really doing two things: creating a dataStatus object and creating a dataStatus test object. This introduces room for bugs if future functions aren't properly assigned to both the real and test object. To catch these bugs, you'd want to write a test case that ensures the test object shares all the same functions as the real object. (i.e. call could have extra functions but it must also have each one that a normal dataStatus would)

I'd recommend eliminating call and using this instead. That way you can do your testing by disecting the methods you want from the real object.

var realObj = new dataStatus (options);
var testObj = {
    setStatus : realObj.setStatus,
    getStatus : function () {},
    signal : function () {}
};

Private functions should be tested through the testing of public functions, but your only private function is for changing functions when testing.

Spelling

status, like sinus, has an -es plural, not an -i - hence, the plural you want for your statii variable is statuses.

Spacing

Be consistent in the whitespace you use. In some methods, you're using the pattern on padding whitespace on both sides:

var methodName = function( argument1, argument2 )

In at least one, you've missed the padding and have this instead:

var methodName = function(argument1, argument2)

I also recommend padding space around else: }else{ should be } else { for readability.

It's a minor point, but consistency in your minor points makes consistency in the major points more natural.

Naming

Generally good variable naming, but beware that local is a reserved keyword in ES6.

Equality

JavaScript has two equality checks: == and ===. I'll spare you the extensive and weird details of each of them, but suffice to say that double-equals is rough equality (i.e. 0 == false) and triple-equals is exact equality.

You should use === when comparing with strings, booleans, undefined or null.

Attribution

It's good that you're attributing where your code came from with that comment hyperlink to SO. However, you also need to include and hyperlink the author's name.