Question

Why thermal conductivity of metal higher than that of fluid?

Answer 1

In solids and liquids, heat can be transported by both atomic lattice vibrations (specifically, acoustic phonons on a crystalline solid, which generalize to sound waves in a liquid) or free electrons.  In addition, liquids can support convective heat transfer.

Liquids have lower thermal conductivity than solids because at the atomic scale, they are basically disordered solids.  Disordered solids also have poorer thermal conductivity than solids with a high degree of crystallinity because acoustic phonons are scattered at imperfections such as lattice dislocations, impurities, and grain boundaries.  Metals tend to have higher thermal conductivity than non-metals (try putting your hot coffee in a copper mug) because the former has free electrons to carry heat and the latter does not.

For the specific case in the question, glass vs mercury, the glass in the question likely does not have a high degree of crystallinity at the atomic scale, which limits its lattice thermal conductivity.  In addition, it is an insulator (and a pretty good one if it is transparent), which limits its electronic thermal conductivity