This is probably the resullt of "chloramination" by your water treatment facility. In the procedure, chlorine and ammonia are added to the water and allowed to react, forming chloramine. This is not an ideal situation, but it is easily managed. Like Chompers said, when you dechlorinate your municipal water, use either Prime or Amquel+. Either of these will detoxify the incoming ammonia and tie it up until your biofilter can oxidize it to nitrite. I would suggest that you perhaps increase the dose of either dechlorinating reagent by 25-50%. I would do nothing more than that. As previously stated, most test kits will still indicate the presence of "free ammonia", but it's actually bound and detoxified.
Finally, since you are getting identical readings between your tap and tank, you might want to try retesting with a different kit.