Why don't shipping businesses own their own shipping containers?
First, it appears there are historical reasons: it was the initiative of the producers to standardize the way their goods were transported, already from the factory. And it involved transportation by rail initially, rather than ship. See interesting information here.

Second, I see an economic rationale: overseas commerce is rather volatile, especially at individual-shipping-company level. In such a situation, it would be difficult for each individual shipping company to optimize the number of containers owned. But if there are companies that lease out containers to all the shipping industry, fluctuations dampen and overall capacity is more easily optimized.

Another approach can be found here, where on the subject the authors write


  “Own or lease?” The question has long lain at the heart of container-shipping strategy. From our analysis, the industry typically
  relies too much on leasing. While leasing may be the only option for
  many cash-strapped liners that already have substantial debt, other
  lines should take advantage of this by owning more of their fleet.
  Leasing does provide a little more flexibility to change vessel
  deployment. But that breathing room often comes at too high a price.


The "substantial debt" usually comes from building/buying new ships.