NaOH may have this effect but it's not the only additive that can have this effect. Remember it doesn't take much NaOH to raise the pH from 7 to 8 (which really isn't THAT drastic from a non-biological perspective). Fresh addition of NaOH to pure water, for example, can change the pH quickly. Think of aqueous solution of NaOH. If you make a 0.1 molar aqueous solution of NaOH, the pH will be 13 or so. You leave that 0.1M solution bottle open to the air, I can guarantee it won't be 13 for long. NaOH will most certainly react with dissolved CO2 (captured from air dissolved in aging water) and one ends up with less and less NaOH over time in an aged solution that is open to the air. In this case, the "solution" just happens to be your tap water.
Remember aging water is not just a single process of "degassing CO2" and driving out gaseous chloride. It is but one or two of the many chemical events happening. While most people will notice a rise in pH with aging water, some will not. I find it helpful to understand principles behind general rules. I think general rules are very helpful and often offer good guidance, but if one doesn't understand the underpinning concept, one cannot formulate a reasoned hypothesis to explain contradictory findings to general rules (or how to handle observations that contradict a general "rule-of-thumb").