Your regex (student\/.*[^\/]*) matches and captures into Group 1 a literal sequence student/, then matches any characters other than a newline, 0 or more occurrences (.*) - that can match the whole line at once! - and then 0 or more characters other than /. It does not work because of .*. Also, a capturing group should be moved to the [^\/]* pattern.
You can use the following regex and grab Group 1 value:
student\/([^\/]*)
See regex demo
The regex matches student/ literally, and then matches and captures into Group 1 zero or more symbols other than /.
Alternatively, if you want to avoid using capturing, and assuming that the ID is always numeric and is followed by /tests/, you can use the following regex:
\d+(?=\/tests\/)
The \d+ matches 1 or more digits, and (?=\/tests\/) checks if right after the digits there is a /tests/ character sequence.
var re = /student\/([^\/]*)/;
var str = 'https://api.bestschool.com/student/1102003120009/tests/json';
var m = str.match(re);
if (m !== null) {
document.getElementById("r").innerHTML = "First method : " + m[1] + "<br/>";
}
var m2 = str.match(/\d+(?=\/tests\/)/);
if (m2 !== null) {
document.getElementById("r").innerHTML += "Second method: " + m2;
}
<div id="r"/>