About sandwich ELISA
This page from Cole-Parmer includes everything you could possibly want to know about the adsorption of biomolecules to polystyrene. To summarize: 


binding is neither covalent nor ionic, but
it is generally mediated by non-specific intermolecular van der Waals interactions, which generally are about 100X weaker than ionic or covalent bonds
the exception is some hydrogen bonds, which can be 10X stronger than the other types

  Intermolecular attraction forces are based on intramolecular electric polarities of which two types can be distinguished: alternating polarities (AP) and stationary polarities (SP), i.e. dipoles.

these interactions can be both hydrophobic and hydrophilic, depending on the surface chemistry of the polystyrene


If you're interested in the specific chemistry of everything, feel free to read the entire thing, it's quite detailed.

I would to note one thing from your question. In the absence of a specific binding partner like Protein A, antibodies bind to polystyrene completely non-specifically, meaning that not only does the Fc region bind, but Fab as well, including the CDRs (antigen binding domains). An ELISA depends on the fact that not all antibody molecules bind this way, so many or a majority of them have their antigen binding sites available to capture their target.