3 digit PIN The SECOND and THIRD digits make a 2-digit number
The FIRST number is a FACTOR of the 2-digit number
The SUM of the 3 digits is 13
The three digit number is a square number

let's start with the last to clues and make a list of all 3-digit square numbers (and by making I mean just looking it up. it would be possible, but it's faster and lazier)

100

121

144

169

196

225

256

289

324

361

400

441

484

529

576

625

676

729

784

841

900

961

(source: https://math.tools/table/square-numbers-list

but I had to copy by hand...)

now let's eliminate all numbers from the list that don't have 13 as the sum of their digits and just keep the few that will remain.

256

625

841

now that's pretty much narrowing it down already, now let's test for the first two clues (I would have no idea how to start with these so I switched the order to make it doable)

The SECOND and THIRD digits make a 2-digit number

The FIRST number is a FACTOR of the 2-digit number

only 256 works, 2 is a factor of 6, but that wouldn't word for the other candidates. so 256 is the number we are looking for.

summing up the second and the third digit (I guess summing up the digits was meant here) also works for 256, if we wanna see a two-digit number as a result.

the key part here was to ignore the order the clues where listed I guess. much easier with just 3 possible numbers.