I can request for permission to write to external sdcard and write any file in that directory.
Not really.
On Android 4.4+, you can use getExternalFilesDirs(), getExternalCacheDirs(), and getExternalMediaDirs(). If those return 2+ items, the second and subsequent ones will be locations on removable storage.
On Android 5.0+, you can use ACTION_OPEN_DOCUMENT_TREE, to ask the user to grant you access to a document tree. That could be the root of removable storage... or anything else the user wants to use. This gives you a Uri back, which you can use with DocumentFile and the rest of the Storage Access Framework. It does not give you filesystem access (e.g., via File).
On Android 7.0+, you can use StorageManager and StorageVolume to ask for access to a removable storage volume. Once again, though, you get a Uri back to use with DocumentFile, ContentResolver, etc -- you do not get filesystem access.