The fact that everything works correctly through the clipboard does not mean that the encoding is correct. The clipboard complements several other formats to the input format (reserves all possible cases).
Your collation on database table is set to UTF8, and you try pass the Unicode text (UTF16).
I checked: do shell script works with UTF16 representation fine. So, the problem is the indicated above.
You should convert polish diacritic texts to UTF8, and only then pass them to your database:
set var1 to "ĄĘŚĆŃŻŹŁ" as «class utf8»
set var2 to "Zażólcić gęślą jaźń" as «class utf8»
do shell script "/Applications/MAMP/Library/bin/mysql " & ¬
"--host=localhost -uroot -proot -D nanc_pusta01 -e \"" & ¬
"insert into articlesTable values(1224420, '2020-11-16 00:00:01', " & quoted form of var1 & ¬
", 1, " & quoted form of var2 & ", 1632, 14, NULL, NULL, NULL, 0, '', '', 0, 0)\""