Similar movies released at the same time

Iron Eagle (January 17, 1986)
Top Gun (May 16, 1986)

Hotshot fighter pilots.

Like Father Like Son (1987)
Vice Versa (March 11, 1988)
18 Again! (April 8, 1988)

Father/son body swaps.

Honorable mention:
Big (June 3, 1988)
A boy finds himself in his own adult body.

Deep Star Six (January 4, 1989)
Leviathan (March 17, 1989)
The Abyss (August 9, 1989)
The Evil Below (1989)
Lords of the Deep (1989)
The Rift (AKA Endless Descent) (1989)

Underwater exploration and strange, mostly hostile creatures.

K-9 (August 28, 1989)
Turner & Hooch (July 28, 1989)

Men team up with lovable dogs to fight crime. Both dogs get shot.

Robin Hood: Prince of Thieves (June 14, 1991)
Robin Hood (TV movie; 1991)

Various adventures of Robin Hood.

Christopher Columbus: The Discovery (August 21, 1992)
1492: Conquest of Paradise (October 9, 1992)
The Magic Voyage (April 9, 1992)
Carry On Columbus (1992)

Various accounts of Christopher Columbus' voyage. This entry might not count, because 1992 was the 500th anniversary of the voyage, therefore these were not coincidental.

Tombstone (December 25, 1993)
Wyatt Earp (June 24, 1994)

The life of Wyatt Earp, including the gunfight at the OK Corral.

Call of the Wild (TV movie; April 24, 1993)
Iron Will (January 14, 1994)
White Fang 2: Myth of the White Wolf (April 15, 1994)

"Sled dog" movies. Iron Will and White Fang 2 were both distributed by Disney. Techcnically, White Fang (the dog) isn't a sled dog, but the time period and setting, as well as the man/boy and his dog(s) story is very similar in all three. Also, both Call of the Wild and the first White Fang were based on books by Jack London.

Honorable mention:
Balto (December 22, 1995)

Gordy (May 12, 1995)
Babe (August 4, 1995)

Heroic talking pigs.

Rob Roy (April 7, 1995)
Braveheart (May 24, 1995)

Historic tales of Scotsmen taking revenge on the Englishmen that killed their wives, in medieval England.

Dantes Peak (February 7, 1997)
Volcano (August 25, 1997)

Large, dangerous volcanic eruptions.

Antz (October 2, 1998)
A Bug's Life (November 25, 1998)

Computer animated comedy/family films about ant colonies. Antz was involved in a dispute between Disney and Dreamworks. You can read more about it HERE

Deep Impact (May 8, 1998)
Armageddon (July 1, 1998)

Large space rocks endanger the Earth while people try to stop them.

Saving Private Ryan (July 24, 1998)
The Thin Red Line (December 25, 1998)

Movies about the horrors of war, set during WWII.

Dark City (February 27, 1998)
The Matrix (March 31, 1999)
The Thirteenth Floor (April 16, 1999)
eXistenZ (April 23, 1999)

Dark futuristic films dealing with virtual reality and existentialism.

The Truman Show (June 5, 1998)
EDtv (March 26, 1999)

A man's life is filmed and broadcast 24/7.

The Sixth Sense (August 2, 1999)
Stir of Echoes (September 10, 1999)

Psychological supernatural thrillers about people who can see ghosts. The music for both movies was written by James Newton Howard.

Mission to Mars (March 10, 2000)
Red Planet (November 6, 2000)
Ghosts of Mars (August 24, 2001)

Astronauts encounter strange things on Mars.

The Score (July 13, 2001)
Heist (November 9, 2001)

An old thief reluctantly teams up with a younger thief for one last job.

Chasing Liberty (January 9, 2004)
First Daughter (September 24, 2004)

Rebellious teenage daughter of the President falls in love with a secret service agent. The working title of Chasing Liberty was First Daughter.

National Treasure (November 19, 2004)
Sahara (April 6, 2005)

A man, a woman, and a comedic sidekick go on a treasure hunt and solve a mystery. Sahara dealt with a Civil War artifact (an ironclad), where National Treasure dealt more with things in the Revolutionary War/Declaration of Independence era, while National Treasure 2 dealt more with the Civil War.

The Illusionist (August 18, 2006)
The Prestige (October 20, 2006)

Time period pieces about seemingly supernatural rival magicians. The Prestige was based on a 1995 novel of the same name. There was a 1995 novel called The Illusionist, but it was unrelated to that movie. Side note: the lead actors in both movies (Hugh Jackman, Christian Bale, Edward Norton) have all played superheroes (Wolverine, Batman, Hulk).

Battle for Terra (August 8, 2007 - May 1, 2009)
Avatar (December 10, 2009)

Good human saves peaceful alien civilization from militaristic humans trying to destroy them. Technically this one might not count because Battle for Terra was released at film festivals in 2007, but it's wide release was in 2009.

Paul Blart: Mall Cop (January 16, 2009)
Observe and Report (April 10, 2009)

Misfit mall cops fight criminals.

Despicable Me (July 9, 2010)
Megamind (November 5, 2010)

Computer animated comedic supervillains and their rivals.

Meteor Man (1993), Blankman (1994)
Prefontaine (January 24, 1997), Without Limits (September 11, 1998)
Finding Nemo (2003), Shark Tale (2004)
Capote (2005), Infamous (2006)
Madagascar (2005), The Wild (2006)
Sky High (2005), Zoom (2006)
March of the Penguins (2005), Happy Feet (2006), Surf's Up (2007)
Firehouse Dog (2007), Bolt (2008)

Two of the articles I used for research on this also shared some similarities with each other. Both of them are Top 11 lists, and both were written between the time when Paul Blart and Observe and Report were released.

11 PAIRS OF SUSPICIOUSLY SIMILAR MOVIES
11 Identical Movies Released at the Same Time