The way class field syntax was designed, such a line can do one thing: it assigns to a property of the instance.

this.name = "tom" isn't syntactically permitted because it would have been redundant.

var name = and other variants aren't permitted in that position because those are variable declaration keywords (var, let, const) - which keyword you use has various implications on the scope of the newly created variable, and whether it can be reassigned. But this makes no sense in the context of a class field, because you aren't creating a new identifier, you're assigning to a property of an object.

This

class User{
    name="tom"

is the same as doing

class User{
    constructor() {
        this.name = "tom";
    }

name is not a new variable created in a particular scope - it's only a property, so using var, let, or const when declaring it would mix two entirely separate concepts, and so is forbidden.

So, you can only do

class User{
    name="tom"

(Unless you want the property to be private (only useable from the inside of the class), in which case prefix the property name with a #.)

class User{
    #name="tom"

I think what you wanted in your question is to know another way to declare variables in a js class. So, you can use a constructor to achieve that, something like that:

class User{
    constructor() {
        this.name = "tom"
    }
 
    showName(){
        console.log(this.name)
    }
}
const tom = new User()
console.log(tom.name)

That’ll print "tom" as well.