"Ama asztal" is kind of archaical with similar meaning, sounds a bit better for me (just try to imagine that 'azaz az az az asztal, ...' is a valid expression) . You can also mention here eme-ama, ezen-azon + ancient Hungarian language did not have a definite article, so 'az asztal' probably meant 'that table' and 'e(z) asztal' probably meant (e still means) 'this table'.
Az is the demonstrative pronoun "that", and a is the definite article "the". The problem is, when the next word starts with a vowel, a becomes az (like "a" becomes "an" in English in front of vowel sounds). So you sometimes have that apparent doubling.
Even more problematic, in front of certain words, the demonstrative az can be reduced to a. :´)
Particles are weird.