Are all monomers in a polymer covalently bonded?
I think your second quote is referring to some polymers such as sodium polyacrylate, $\ce{[-CH2-CH(COONa)$-$]_n}$. This polymer is basically just the sodium salt of polyacrylic acid (or polymerized sodium acrylate). You can see below the ionic bond between the anionic carboxylate group and the sodium cation:



However, the quote could be referring to so-called 'chain-link' polymers, in which the monomers are intertwined and looped/knotted together, like links in a chain, instead of just consisting of a bunch of long strings of monomers, like a rope. However, chain-link polymers are highly uncommon and rather difficult to synthesize. Note that these are not the same as cross-linked polymers.