The answer from Mateusz Lazaru explains what is happening in MDB5 v7.1.0. This is an odd decision by MDB for button colors.

To fix this, here is what I've done so far. Hopefully this is useful for anyone else wanting to use the SCSS variables for $secondary, $light, and $dark for buttons. I still need to get shadows working right but this at least gets colors working. Unfortunately, more work will be needed to get shadows working on .btn-secondary.

Edit: The reason .btn-secondary doesn't get a shadow is because in MDB's _buttons.scss, the attribute for box-shadow is hard coded to none.

// Fixes MDB's use of color variables to set secondary buttons.
.btn-secondary {

    @include button-variant(
        $secondary,
        $secondary,
        color-contrast($secondary),
        $btn-contextual-box-shadow #ff0000 // Not working.
    );
}

// Fixes MDB's use of color variables to set light buttons.
.btn-light {

    @include button-variant(
        $light,
        $light,
        color-contrast($light),
        $btn-contextual-box-shadow shade-color($light, $btn-hover-bg-tint-amount)
    );
}

// Fixes MDB's use of color variables to set dark buttons.
.btn-dark {

    @include button-variant(
        $dark,
        $dark,
        color-contrast($dark),
        $btn-contextual-box-shadow tint-color($dark, $btn-hover-bg-tint-amount)
    );
}