What is the story of Bharat Mata?
What's her connection with the Indian national flag?
The Bharat mata is [primarily] goddess Durga. However in general she is an amalgamation of various goddesses.
Bharat Matā (Hindi, from Sanskrit Bhāratāmbā भारताम्बा; अम्बा ambā means 'mother') is the national personification of India as a mother goddess. She is an amalgam of all the goddesses of Indian culture and more significantly of goddess Durga. She is usually depicted as a woman clad in a saffron sari holding the Indian national flag, and sometimes accompanied by a lion... The Tiranga flag was also started being included during this period. In 1930s, the image entered in religious practice. The Bharat Mata temple was built in Benaras in 1936 by Shiv Prashad Gupt and was inaugurated by Mahatma Gandhi. ... The motto Bharat Mata ki Jai ("Victory for Mother India") is used by the Indian Army. [Wikipedia]
This is a rare combination of religion and nationalism (a part of Dharma).
It was actually a painting work, which was named as "Bharat Mata". Eventually it got transformed into a worshippable figure for indpendence movement of India. The very early history of Bharat mata dates back to 1905 from a painting of Abanindranath Tagore:
Bharat Mata depicts a saffron clad woman, holding a book, sheaves of paddy, a piece of white cloth and a garland in her four hands. The painting holds historical significance as it is one of the earliest visualizations of Bharat Mata, or "Mother India." ... The "work" was painted during the time of the swadeshi movmement. The movement began as a response to the Partition of Bengal (1905) [source]
She is not a traditional Pauranic goddess.
So in the oldest of paintings, there was neither tricolor flag with her, nor was there any lion. The flag started to appear around 1930's when the independence movement was at its peak as described in the above Wiki article.
Here is what Sister Nivedita, a famous student of Vivekananda has to say:
From beginning to end, the picture is an appeal, in the Indian language, to the Indian heart. It is the first great masterpiece in a new style. I would reprint- it, if I could, by tens of thousands, and scatter it broadcast over the land, till there was not a peasant's cottage, or a craftman's hut, between Kedar Nath and Cape Comorin, that had not this presentment of Bharat-Mata somewhere on its walls. Over and over again, as one looks into its qualities, one is struck by the purity and delicacy of the personality portrayed. [Complete works of Sister Nivedita]
As we know Bhumi is considered as Mata according to scriptures.
माता भूमि: पुत्रोऽहं पृथिव्या: (Atharva Veda 12.1.12)
Now, talking about Bharat Mata, there is history of freedom movement and bit nationalism is connected with it.
Kiran Chandra Bandyopadhyay first coined the term "Bharat Mata" as the title of one play/drama in 1873.
In 1882 Bankim Chandra Chattopadhyay's novel Anandamath introduced the hymn "Vande Mataram" which soon became the song of the emerging freedom movement in India.
In which he called Bharat Mata as Lord and equated her with goddess like Durga, Laxmi etc.
त्वं हि दुर्गा दशप्रहरणधारिणी
कमला कमलदल विहारिणी
वाणी विद्यादायिनी, नमामि त्वाम्
नमामि कमलां अमलां अतुलाम्
सुजलां सुफलां मातरम् ॥ वन्दे मातरम्
In 1905, Abanindranath Tagore portrayed Bharat Mata as a four-armed Hindu goddess wearing saffron colored robes, holding a book, sheaves of rice, a mala, and a white cloth.
In 1936, a Bharat Mata temple was built in Benaras by Shiv Prashad Gupt and was inaugurated by Mohandas Gandhi.
- Two another temples at Haridwar in 1983 and at Kolkata in 2015 are founded. In the Kolkata temple, Bharat Mata is portrayed through the image of Jagattarini Durga.
So, Bharat Mata is not goddess described in scriptures but the consideration of Bharat Bhumi as Mata i.e Bharat Mata is portrayed looking like goddess.
Thus, the practice of worshiping Bhumi as Mata is in accordance with scripture whereas consideration of Bharat Mata as goddess Durga is feigned (the word choice may be not appropriate).
Reference for historical points: Vivekananda Kendra.
We all know there is no such lady like Bharat Mata.
It is just the creativity of an artist.
I can show you many examples on the Internet, where Bharat Mata stand with Bhagwa flag.
Seems clear eye creativity that shows a sign to be proud on our country and give respect to it as a Mother and visualize the importance of Tiranga Flag.
Bharat mata is a personification of the nation that is India. No mythological records exist in the Puranas or Itihasas. It was created by the nationalist leaders to inspire the national movement during the late 1890s. And as for the flag, the Indian Tricolor is a modified version of the pre-1947 official flag of the Indian National Congress. Originally the flag that Bharat mata was portrayed to hold, was the INC flag.
The Indian Culture could easily identify the Country with the Mother Goddess, because, we had in Sri Sri Chandi(chapter 11, mantra 4).
aadhaarabhutaa jagatas tvamekaa mahisvarupena yatah stitaasi
O Divine Mother! You are the resort of all as you are present in form of land.
We also had in our culture
janani janmabhumish cha svargaad api gariyasi Mother and Motherland are more glorious than heaven.
So Bharat-Mata was seen as a form of the Mother Goddess. The Indian Tri-colour flag was introduced after independence I think.