What is used to de-ice a commercial aircraft?
De-icing fluids are used which are made of ethylene glycol (EG) or propylene glycol (PG) in addition to other ingredients.

What is used to de-ice a runway at a large airport?
Airports use different types of materials, e.g. potassium acetate. Safegrip ECO2 is a brand which is popular.
Runways can also be deiced using normal techniques like salt and snowplows etc.
Do the two fluids effect each other at all? Meaning, when an aircraft drips de-icing fluid onto the runway, does it cause any interactions or problems?
I do not think that there is much or any damage to a runway from the de-icing fluid. If something is bad for solid concrete, how it can be good for fragile aluminium?
Research shows that the residuals left on the airplanes cause some issues. Those are described below.
Effect of runway de-icing fluid residuals on airplanes
There are some studies that show the runway fluid having some adverse effects when combined with de-icing fluid residuals on airplanes. They state:
Runway deicing fluids contain potassium acetateor potassium
formate-based fluids (deionizing salts). When these fluids combine,
the separation of the anti-icing fluid thickeners may be enhanced,
leading to a more rapid formation of the residue.
Effect of airplane de-icing fluid residuals on airplanes
However, de-icing fluid residuals on airplanes can freeze and cause damages to flight controls. More information is here.
