From the standard’s introduction (3rd edition, 1st September 2011):

“An ISIL identifies an organization, i.e. a library, an archive, a museum or a related organization, or one of its subordinate units, which is responsible for an action or service in an informational environment (e.g. creation of machine-readable information). It can be used to identify the originator or holder of a resource (e.g. library material or a collection in an archive). The ISIL is intended for use by libraries, archives, museums and agencies doing business or interacting with these organizations (e.g. suppliers, publishers, and government institutions). An ISIL identifies an organization or one of its subordinate units throughout its life. ...”

Accordingly any organisation (including non-library institutions) could obtain an ISIL if needed for applications in information technology.

ISIL is being supervised by the ISIL Registration Authority in Copenhagen. 

The German ISIL Agency was established in 2004 and has been assigning ISIL for German organisations since then. The German ISIL can be retrieved from the ISIL database.

ISIL for libraries, archives and related organisations
ISIL for libraries and archives will be assigned according to already existing library codes. The ISIL consists of a country code and a consecutive number.

Examples:

DE-1 = Staatsbibliothek zu Berlin - Preußischer Kulturbesitz, Haus Unter den Linden
DE-700 = Universitätsbibliothek Osnabrück
DE-1900 = Stadtbibliothek Leuna
DE-1939 = Norddeutsches Nah- und Schienenverkehrsarchiv, Ahrensburg

Normally ISIL are being assigned as consecutive numbers. However, alphabetic characters may also occur if already existing library codes were used to create the ISIL.

ISIL for museums
The ISIL for German museums will be assigned by the Institut für Museumsforschung der Staatlichen Museen zu Berlin (IfM) by using hitherto existing museum numbers. The relevant data of museums for the museums will be imported into the ISIL database. By publication within this directory the ISIL of the museums become validated ISIL. All ISIL for German museums start with the prefix "DE-" being followed by the sub-prefix "MUS-" which is then followed by the number of the museum.

Examples:

DE-MUS-047727 = Saarländisches Bäckereimuseum, Ottweiler
DE-MUS-296617 = Stadtmuseum Hagen
DE-MUS-319517 = Deutsches Historisches Museum, Berlin

ISIL with the prefix "ZDB-..."
These are mainly being used for packages of digital resources regarded as virtual libraries. The ISIL facilitates metadata transfer between library networks and associated libraries (s. ISIL for products).