Why do we say that macrophages are part of the innate immune system if they still have to recognise the antigen with specific receptors like the T cells and B cells?

What is exactly the role of antibodies? I know they activate compliment proteins but am I correct to think that they make it easier got eosinophils, monocytes and neutrophils to phagocytose them?

That's part of their adaptive immunity ability. They are very versatile little guys.

But why do we classify them as innate?

Antibodies identify and neutralize foreign objects such as bacteria and viruses for the immune system.

The immune system is innate.

From Wikipedia --

innate immune system = non-specific immune system, first line of defense, comprises the cells and mechanisms that defend the host from infection by other organisms in a non-specific manner. This means that the cells of the innate system recognize and respond to pathogens in a generic way, but unlike the adaptive immune system, it does not confer long-lasting or protective immunity to the host.