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You can use .test() on your regular expression to check whether the pattern matches part of the string. As the pattern will need to match only a portion of the string, you will need to remove the ^ and $ characters as your matched pattern can be contained within a line. Also, there is no need to escape the character class bracket as you have tried to do so in your expression (\[), as you want the [ to be treated as a character class, not a literal square bracket:
const myString = "This is my test string: AA.12.B.12 with some other chars";
const res = /[A-Z]{2}\.\d{2}\.[A-Z]{1}\.\d{2,3}/.test(myString);
console.log(res);
If we look at the docs for String.prototype.includes() we can see it takes a string, not a regular expression. Instead, try RegExp.prototype.test().
Also, since you only want to match a sub-string, you'll need to remove the start and end anchors ^ and $. (More on regex anchors)
Your code becomes:
var myString = "This is my test string: AA.12.B.12 with some other chars";
var myRegex = /[A-Z]{2}\.\d{2}\.\[A-Z]{1}\.\d{2,3}/;
console.log(myRegex.test(myString));