Why is the amino acid cysteine classified as polar?
[...] we still put it in polar category

If you search for amino acid classification, you will find that there is no agreement on how to classify cysteine.
What you should know is that cysteine has different roles in proteins. In cysteine proteases, it acts as a nucleophile (and its surrounding often makes it more prone to deprotonation compared to a netural aqueous environment). Cysteines that form disulfide bonds often are at or near the surface of the protein; once the disulfide is formed, you have a quite hydrophobic group. Protonated cysteine is incapable of making conventional hydrogen bonds, and the electronegativity of carbon and sulfur are quite similar. This explains why methionine, the other sulfur-containing amino acid, is classified as hydrophobic.

[...] , why?

The classification of amino acids is sometimes used as proxy of the tendency of amino acid side chains to be located on the surface or in the interior of a protein. Similarly confusing, proline is sometimes classified as polar amino acid; however, its side chain is made exclusively of carbon and hydrogen atoms. If you study where proline is mostly located with a folded protein, you find it is on the surface. This is not because proline is polar but because it is a helix and strand &quot;breaker&quot;, so it is often found in turns, which are mostly on the surface of the protein.
Also, you should know that there is no all-or-nothing classification of every amino acid side chain. Tyrosine, threonine, even lysine have hydrophobic parts while being capable of making hydrogen bonds. Often, the environment is just right, with hydrophobic parts interacting with other hydrophobic parts, and all hydrogen bond potential satisfied, either by interactions with other side chains, with main chain or with solvent.
Why is cysteine not classified in a more consistent manner?
Here is an excellent starting point to answer this question, starting with the abstract:

Cysteine (Cys) is an enigmatic amino acid residue. Although one of the least abundant, it often occurs in functional sites of proteins. Whereas free Cys is a polar amino acid, Cys in proteins is often buried and its classification on the hydrophobicity scale is ambiguous. We hypothesized that deviation of Cys residues from the properties of a free amino acid is due to their reactivity and addressed this possibility by examining Cys in large protein structure datasets. Compared to other amino acids, Cys was characterized by the most extreme conservation pattern, with the majority of Cys being either highly conserved or poorly conserved. In addition, clustering of Cys with another Cys residue was associated with high conservation, whereas exposure of Cys on protein surface with low conservation. Moreover, although clustered Cys behaved as polar residues, isolated Cys was the most buried residue of all, in disagreement with known physico-chemical properties of Cys. Thus, anomalous hydrophobic behavior and conservation pattern of Cys can be explained by elimination, during evolution, of isolated Cys from protein surface and clustering of other Cys residues. These findings indicate that Cys abundance is governed by Cys function in protein rather than by the sheer chemical and physical properties of the free amino acid, and suggest that high tendency of Cys to be functionally active can considerably limit its abundance on protein surface.