Is there any other kingdom than Hyrule?

Yes, there are places other than Hyrule kingdom.

Koholint Island

An island which Link visits during the events of Link's Awakening and is quite separate from Hyrule. While there is a castle on Koholint Island, it belongs to a nobleman, and not to a king.

The Sacred Realm / Dark World

First introduced in A Link to the Past, the Dark World exists in a dimension parallel to Hyrule and is a corrupted version of the Sacred Realm (where the Triforce rests). Despite being called a "realm", it does not seem to have a ruler, and therefore cannot really be called a kingdom.

Termina

Termina is where Majora's Mask takes place. It is considered a parallel world to Hyrule due to its stark resemblance to the Kingdom. Termina is split in 5 areas: the swamps, home to the Deku Scrubs and their king; the mountains, home to the Gorons and their elder; the ocean, where the Zoras live; the Ikana Canyon, where the Ikana kingdom once stood; and Clock Town, which is ruled by a mayor.

Labrynna

The setting for Oracle of Ages, it is one of few places that aren't a parallel version of Hyrule one way or another. While Labrynna is no longer a kingdom, it used to be 400 years before the events of the game. Link can actually visit Labrynna kingdom by travelling to the past using the Harp of Ages.

Holodrum

Where Oracle of Seasons takes place. Similar to Labrynna, it is another place that is no parallel version of Hyrule. Holodrum is mostly uninhabited. Below Holodrum is Subrosia, whose Subrosian population does not seem to have a ruler.

The Twilight Realm

Twilight Princess' parallel version of Hyrule is home to the descendants of a people that have been banned from Hyrule a long time ago. Ruled by a Twili royal family, it can be considered a kingdom.

Skyloft

While floating above what will eventually be known as Hyrule, it is difficult to say if Skyward Sword's starting town does belong to Hyrule kingdom or not. Skyloft isn't ruled by a king, however.

Lorule

Often confused with the Dark World, Lorule is not A Link Between Worlds' version of the Dark World. It is instead yet another parallel version of Hyrule. Ruled by its own version of Hyrule's royal family with its own "Princess Hilda", it is as much a kingdom as Hyrule is.

Hytopia

Since I haven't played Tri Force Heroes, I do not know the relationship between Hyrule and Hytopia. All I know, is that they aren't the same place. Hytopia is currently ruled by King Tuft, whose daughter is called Princess Styla.