"Ionized water" is nothing more than sales fiction; the term is meaningless to chemists.
Pure water (that is, water containing no dissolved ions) is too unconductive to undergo signficant electrolysis by "water ionizer" devices.
Pure water can never be alkaline or acidic, nor can it be made so by electrolysis. Alkaline water must contain metallic ions of some kind — most commonly, sodium, calcium or magnesium.
The idea that one must consume alkaline water to neutralize the effects of acidic foods is ridiculous; we get rid of excess acid by exhaling carbon dioxide.
If you do drink alkaline water, its alkalinity is quickly removed by the highly acidic gastric fluid in the stomach.
Uptake of water occurs mainly in the intestine, not in the stomach. But when stomach contents enter the intestine, they are neutralized and made alkaline by the pancreatic secretions — so all the water you drink eventually becomes alkaline anyway.