How to select element that contains only the class of “foobar”, don’t select if it contains any other classes?

The simplest way with pure CSS is to use an attribute selector:

[class="foobar"] {
    /* only apply to an element if the element has a class and the only class is 'foobar'. */
    background: black;
    color: white;
}

You may not have control over the other elements, but if you have control over yours and you can modify its class attribute, mayabelle makes a good point about assigning your element its own special class and selecting by that class instead. That way you explicitly state which element is yours rather than going by the assumption that not having additional classes means it’s yours.