Is the "index.js" module resolution (specifying the containing folder) supported by the ES6 (ECMAScript 2015) modules official standard?
No. ES2015 doesn't contain anything about the module loader or how to interpret module identifiers.
Or is it just "custom" Node.js/CommonJS transpiling behaviour?
Yes. You can read about the module resolution algorithm in the documentation. The relevant part:
require(X) from module at path Y
1. If X is a core module,
a. return the core module
b. STOP
2. If X begins with './' or '/' or '../'
a. LOAD_AS_FILE(Y + X)
b. LOAD_AS_DIRECTORY(Y + X)
3. LOAD_NODE_MODULES(X, dirname(Y))
4. THROW "not found"
[...]
LOAD_AS_DIRECTORY(X)
1. If X/package.json is a file,
a. Parse X/package.json, and look for "main" field.
b. let M = X + (json main field)
c. LOAD_AS_FILE(M)
2. If X/index.js is a file, load X/index.js as JavaScript text. STOP
3. If X/index.json is a file, parse X/index.json to a JavaScript object. STOP
4. If X/index.node is a file, load X/index.node as binary addon. STOP
Will it be possible to omit the /index|/index.js part of the import in all browsers as well (when modules will be supported on all browsers)?
Hopefully browser implementations will aim for maximum compatibility with existing module loaders, but for now we don't know. Maybe it doesn't have anything to do with the browser either, but with how the server resolves module identifiers. I confess that I haven't followed the development lately, so any other insights are much appreciated :)