N#168265 or N#168272 - both fit. Both mints have same AT mark in 1665 according to Numista. If possible add first image same way as you added second one as it got mark below portrait. You added first image in a way that it can't be enlarged.
It is not Lithuanian 6 groschen as reverse lettering would be different.
Your other option would be to get Kopicki book and check which number fits.
Main page gives you lettering on main image. Period lettering consistency was not consistent. :) Numista would need to double their database storage capacity to keep all lettering variations from 1500-1700.
I would be checking mint marks, year and assume that lettering can be different in some variant. I got only Lithuanian variant source myself. There are 18 Lithuanian lettering variants in 1665.
Your's seems to be the most similar with my coin, but the year is not good. Can you help somehow to find out if I made the year identification wrong or something? Also, I can't find A T on the coin you showed.
Sorry, I know very little about these coins. If there is a S M D L R (Sweden, Grand DUke of Lithuania and Ruthenia) variant on one of the coins tokul posted that would be best but I don't have any references for them.
And one last question, on the Coin Holder should I write Polish-Lithuania 1665 6 Groschen Jan II Kazimir Waza and what mint do you think it is?
If you want English language on holder, ruler is John II Casimir Vasa. Numista's directory for Commonwealth is Polonized. Standard catalogue of world coins reference is KM 91, but they do not cover variations. Andrzej Tymp (AT) fits Krakow in 1665 according to KM.
However … it appears this is one of the most difficult varieties to attribute (or there is some new scholarship) because I can find some older attributions of the coin to Krakow.
Same auction house citing the same reference, which is a specialty book only on the szostaks of this ruler.
The szostaks of Jan Kazimierz/Jan Casimir are hard to attribute because they were struck in very large numbers, meaning there are multiple die varieties for each year at each mint, often with small spelling variants which might or might not be important to the attribution. Even the well-known Polish auction houses do not always seem to be in agreement.