The adapter is stamped 6 A 125V - 250V
I am guessing these products are made in some 3rd world country, and there is no regulation on what they are allowed to claim on them. But in Europe, two pin adapters such as that one are limited by code to 2½ amps, so it would be technically "illegal" to use it for 6 amps, and it should not be recommended for that on this site (per the guidelines).
There’s a mismatch between the adapter and your iron.
No mismatch. Although the manufacturer probably doesn't even know what that means, if he did, it would only mean it is rated up to 250V (the 125V is irrelevant), and it can conduct up to 6 amps. The input for the straightener is within the voltage limits.
BTW, the page for the straightener has a misstatement, it is not dual voltage. Dual voltage means it operates on just two voltages with a ratio of 2:1 (230V and 115V, for example). A switch is used to put the resistive elements, which are designed for the lower voltage, in series for the higher voltage.
This device is multi-voltage. It takes the input voltage, whatever it is, more than 100V, and limits the output electronically to a lower voltage, probably in this case 100V, which is what the resistive elements are designed to work at. This is the way all USB voltage converters (chargers) work.