can two atoms of the same element form ionic bonds, covalent bonds, or both?

Well, when we talk about compounds where are the same elements, it is supposed that the bond between them is covalent. To tell more, it is a covalent non-polar bond (you can see it in N2, O2, Cl2 etc). It means that both atoms (or elements, to be more exact). have the same electronegativity number. So there is no electron shift (it happens when there are atoms with different electronegativity - electrons forming bonds shift to an atom with the biggest electronegativity).

Between the same elements there is no electron shift so this bond is covalent (it is non-polar).

Ionic bond is the limiting case of covalent polar bond. It means that there are two atoms with a big difference between numbers of their electronegativities (good ionic bond appears between a metal and a non-metal).

Such bond is always polar. But it can t appear between the same atoms because their electronegativities are similar (it is impossible to make a positive or a negative ion there).

All such compounds are covalent, but they can r show ionic properties because there is no positive or negative charge - bond is divided between these atoms in equal parts without any electron shifts (it is typical for polar or ionic bonds).

There is no driving force to delocalize electrons from atoms that are the same and form ions, as long as there are no other atoms in the molecule. So, they can only share the electrons in covalent bonds.

I don't know of any cases where 2 atoms of the same element form ionic bonds, but many form covalent bonds. Oxygen forms O2 molecules, so does nitrogen, chlorine, fluorine, and hydrogen. There might be more that I am forgetting.