Where was Lord Murugan born and in which city?

 According to the Sangam text Paripadal, Lord Murugan, who is the son of Shiva and Parvati, was born in the Himalayas.

( The seven wives (all except the chastest Arundhati), the Kārttikai matrons took in the fire-returned foetal pieces without jeopardising their chaste status. The pure wives of the great sages thus carried in their wombs the divine embryo. In the blue, cool tarn in the lofty Himalayas, in six lotuses, the six divine babes were born. Lord Murukan of deathless fame everlasting! On the very day You were born, the Lord of the Devas, Indra of rare power, attacked You with his fire-spitting weapons. Fusing Your six forms into one glorious frame, bare-bodied, weaponless, and yet an infant, You made Indra quail before Your power, despite his mighty Vajrayudha )

Source: paripadal 5, Ettuthokai.

If Lord Murugan is the "Tamil God," do the Telugus, Malayalis, or Kannadigas have their own unique regional god as well?

 Kannada has a regional goddess, she is Bhuvaneshwari Devi.

She is considered to be a form of Parvathi Devi.

Bhuvaneshwari Devi is considered as mother Kannada, she is referred to as Kannada taayi Bhuvaneshwari. In Kannada, taayi means mother.

And this concept exists right from the time of first Kannada empire, Kadambas.

The idea of a temple dedicated to Bhuvaneshwari Devi was conceived by the Kadambas first.

We cannot find her mention during Chalukyas and Rashtrakutas era.

But, it was Vijayanagara rulers who carried forward the legacy.

They built Bhuvaneshwari temple at Hampi, next to the Virupaksha temple, the presiding deity of Vijayanagara empire.

It is the belief the Bhuvaneshwari took the avatar to help revive a Kannada kingdom. She is believed to have helped Hari Raya and Bukka Raya (founders) to establish the new kingdom.

Virupaksha and Bhuvaneshwari temple, Hampi.

Mysore Wodeyars too who were influenced by Vijayanagara rulers built a Bhuvaneshwari temple inside Mysore palace premises.

Bhuvaneshwari Temple, Mysore palace.

This was the idea that was adopted during the fight for the unification of Karnataka movement.

Even now, see any Kannada/Karnataka flag, the devi in that flag is Bhuvaneshwari.

We Kannadigas believe she will protect Kannada language, people and the region forever.

Even the Kannada flag was designed based on turmeric and kunkuma colours which are used for devi worship.

That’s the reason for yellow and red colours in Kannada flag.

You can find Bhuvaneshwari photo in every Kannada flag, be it auto stand or a Kannada sangha.

Apart from Bhuvaneshwari Devi, many Kannadigas consider Anjaneya also as Kannada God as he was born in Karnataka.

As per Ramayana, Hanuman was born in Kishkinda, that is modern Hampi in Karnataka.