For an excellent review of the active site structure of serine proteases, see Branden and Tooze, Introduction to Protein Structure, chapter 15.
The following schematic depicts the active site of papain. Note that a catalytic triad can be formed between Cys25 (the nucleophilic residue), His 159 and Asn 175. The difference between this triad and that found in serine proteases is that the serine is replaced by cysteine and the aspartate is replaced by asparagine. One could hypothesize that the lower pKa of cysteine means that a less basic histidine is required. Perhaps by substituting the uncharged but polar asparagine for aspartate, it is possible to stabilize the correct tautomer of histidine but at the same time lower its pKa.
The nature of the oxyanion hole that is found in papain is actually fairly similar to that found in subtilisin. The amide nitrogen of the nucleophilic residue (Cys 25 here, or Ser 221 in subtilisin) donates a hydrogen bond to the scissile carbonyl oxygen of the substrate. In addition, glutamine 19 donates a second hydrogen bond to the carbonyl oxygen, in a fashion that is analogous to the function of asparagine 155 in subtilisin.
It is unlikely that trypsin, subtilisin or papain arrived at these similar active site geometries through divergent evolution from a common ancestor. Consider the positions of the catalytic triad residues in the primary sequence:
Trypsin: His 57 - Asp 102 - Ser 195
Subtilisin: Asp32 - His 64 - Ser 221
Papain: Cys 25 - His 159 - Asn 175
The recruitment of the three residues from different spots in the primary sequence of the three proteins makes common ancestry to a prototypical serine/cysteine protease unlikely.