In TypeScript, every value except for null and undefined is assignable to the so-called empty object type {}. So you can simplify your Something to just {} and your user-defined type guard function will work the same as before:

export type Something = {};
function isSomething(value: unknown): value is Something {
  return value !== undefined && value !== null;
}
function narrowing(o: unknown) { // OK
  if (isSomething(o)) {
    console.log(o.toString(), o.constructor);
  }
}

Furthermore, TypeScript 4.8 will introduce improved support for narrowing unknown, after which your "NotNarrowing" function will start working, because checking (o !== null && o !== undefined) will narrow o from unknown to {} automatically:

// TS4.8+ 
function nowNarrowing(o: unknown) { // okay
  if (o !== null && o !== undefined) {
    console.log(o.toString(), o.constructor);
  }
}

Playground link to code

@jcalz's excellent answer led me to the following solution:

// With `IsSomething` type guard, I want to narrow down access 
// to common properties of any JS variable 
// which is not null and not undefined, like: 
// .toString, .valueOf, .hasOwnProperty, .constructor etc.

export function IsSomething(value: unknown) : value is Object {  
  return value !== null && value !== undefined;
}

Narrowing([1, 2, 3]);

function Narrowing(o: unknown) { // OK
    if (IsSomething(o)) {
      console.log(o.toString(), o.constructor);
    }
}

It turns out, TypeScript has a predefined interface Object, which works exactly the way I want:

In TypeScript, the Object interface is different from the primitive object type, as the former is still assignable to any other primitive or complex type. It is also different from an empty type like type Something = {}, as the latter doesn't supports IntelliSense.

// Object interface is OK
let a: Object = 42; 
let b: Object = true;
let c: Object = "string";
let d: Object = Symbol(42);
let e: Object = new class SomeClass { someProp = 42 };

// Empty type is OK but no IntelliSense 
let a1: {} = 42; 

// Error: Type 'number' is not assignable to type 'object'
let a2: object = 42; 

Playground link.

One subtle issue still remains. As @CertainPerformance pointed out, the above won't work for an edge case like Object.create(null), where an object is created which doesn't have any properties or methods, but isn't null either.

If that's a requirement, the following should do the job:

// can be extended to check for toString, valueOf, etc
export function IsSomething(value: unknown) : value is Object {
  return (<Object>value)?.constructor !== undefined;
}

Narrowing([1, 2, 3]);

function Narrowing(o: unknown) { // OK
  if (IsSomething(o)) {
    console.log(o.toString(), o.constructor);
  }
}