Can one depressive episode be enough to prescribe antidepressants?
Suppose someone were to see a psychiatrist, explain their life
  problems and such and then their psychiatrist concludes they are
  currently in a depressive episode. Their psychiatrist doesn't yet know
  if they have major depressive disorder - MDD - bipolar disorder or no
  disorder.
  
  Are there situations where someone in such scenario might be able to
  be prescribed antidepressants or any kind of medication?


Yes, absolutely. I do not think that they are necessarily rare either. As this link attests:

The proportion of non-psychiatrist doctor visits where antidepressants were prescribed without a documented psychiatric diagnosis increased from 59.5% to 72.7% between 1996 and 2007, according to a new study in Health Affairs.


  Does the fact that they have been prescribed such medications imply
  that they have been diagnosed with or highly thought by a doctor to
  have a mental illness instead of simply having an "episode" or a
  temporary mental health problem?


Being prescribed the medicine implies that the doctor believed that they were suffering from depression, a form of mental illness. Other factors (e.g., the symptoms or their history) would dictate whether the doctor though it was/would be a long term mental illness/mental health problem.