1) Both begun with the major advancements in AI and Robotics

In the Matrix the Second Renaissance begun when Mankind developed Machines with true Artificial Intelligence, and with Omnics such as Zenyatta, Tekhartha Mondatta and Bastion, I believe it is fair to say that Mankind in the Overwatch universe have achieved the same technological advancements as well.

2) Both Overwatch and our world in the Second Renaissance are set in the near future

The French and Korean version of the cinematic says the Petras Act was signed in 2042.

http://overwatch.wikia.com/wiki/Petras_Act

The Second Renaissance was an era of machine prosperity and human decadence between 2090 and 2139.

http://matrix.wikia.com/wiki/Second_Renaissance

3) Similar technology can be seen throughout both worlds, such as Self Aware AI and Hover cars

Hover car being produced by 01 Heavy Industries

Hover car as seen throughout Volskaya Industries

4) In both realities war was triggered by the sabotage of the Machine’s attempt to achieve peace with Mankind.

The Machines requested admission to the United Nations, presenting plans for a stable, civil relationship with the nations of man. Their admission was denied, and 01 was subjected to a prolonged barrage of nuclear weapons.

http://matrix.wikia.com/wiki/The_Second_Renaissance

The Second Omnic Crisis may have begun with the assassination of the omnic religious leader, Tekhartha Mondatta. He was killed by Widowmaker in King's Row after making a speech espousing peace.

http://overwatch.wikia.com/wiki/Second_Omnic_Crisis

5) Both wars were a brutal struggle of survival between Man and Machine

The Machines advancing forward during the Machine War

The Omnics advancing forward during the Omnic crisis as seen in The Last Bastion

6) Mankind in both realities were failing initially to halt Machine advances, and thus were forced to develop new measures to counter the Machines/Omnics.

a) In the OW universe, Mankind’s new measures granted it victory over the Omnics during the Omnic Crisis and hence ensured its survival and its control over the Omnics.

When the individual self-defence programs proved to be failing, the United Nations developed a small, experimental task force called Overwatch. Its members were among the world's most ingenious soldiers and strategists.

http://overwatch.wikia.com/wiki/Omnic_Crisis#Initial_Response

b) In the Matrix Universe, Mankind’s new measures brought its demise and contributed to its own defeat during the Machine war, bringing with it the near annihilation of the human civilisation and the total subjugation of the human race under the Machines.

Quickly finding themselves losing, man resorted to blocking out the sun, reasoning that as the Machines mainly ran on solar power, they would be hopeless without it. Unfortunately the Machines still soldiered on, running on temporary energy sources, while man's agriculture and ecosystem fell into ruin.

http://matrix.wikia.com/wiki/The_Second_Renaissance

Thus is how the two realities differ, one being the story of Humanity’s triumph over its own creations, whilst the other being the tale of Machine’s triumph over its own creators.

tl;dr: Overwatch and the Matrix are set in the same universe but Overwatch is the alternate reality where Humanity triumphed over the Machines through the creation of the experimental task force called Overwatch as opposed to the disaster known as Operation Dark Cloud which led to the fall of Mankind.