How can an enzyme favor even chained fatty acids?
The first experiment which showed that fatty acids are oxidized in C2-units has been done by Georg Franz Knoop and been published 1904 as "Der Abbau aromatischer Fettsäuren im Tierkörper.". The paper in reference 1 states:


  Georg Franz Knoop discovered fatty acid β-oxidation. In 1904, he
  published his classical experiments using odd and even chain ω-phenyl
  fatty acids such as ω-phenylvaleric acid and ω-phenylbutyric acid
  (Knoop 1904). Knoop fed these compounds to dogs and analysed their
  urine. In dogs that had been fed the odd chain fatty acids, he found
  hippuric acid (conjugate of benzoic acid and glycine), whereas, the
  dogs that had been fed even chain fatty acids excreted phenaceturic
  acid (conjugate of phenylacetic acid and glycine). From this he
  concluded that the metabolism of fatty acids proceeds by the
  successive removal of two carbon fragments. The remaining fatty chain
  had to contain a carboxylic acid. He postulated that oxidation took
  place on the β carbon atom, an oxidation unknown to organic chemistry.


If the fatty acid chain has an uneven number of carbon atoms, the resulting propionyl-CoA is modified into succinyl-CoA which is then used by different processes.

During the breakdown of the chain, Acetyl-CoA groups are released (at the end of each step), which can then be further metabolized by the body. Acetyl-CoA is fed into the Krebs cycle and then oxidized.

References:


A general introduction to the biochemistry of mitochondrial fatty
acid β-oxidation
Fatty Acid Oxidation