You have mentioned that The real name of Hinduism is Sanatan Dharma, the eternal religion. The word hinduism is given by other people due to hindus living beyond the river Sindhu, hence called hindus
You have aleady given more than half of the answer in just two lines
As you have aptly mentioned hindu is the word given by other people to those who lived beyond sindhu river, but I would like to add a point here , this hindu word is applicable only those who followed sanatan dharma , exempting explicitely those who donot follow like Christians, muslims etc but add those who follow the sanathandharma like Jainism and Sikhism are included
“””Etymologically, "Hindu" is simply the Persian form of "Sindhu", meaning generally "river" and specifically "the Indus river", hence meaning "one who lives in (or, by extension, beyond) the Indus river", "an Indian”””
Your first sentence The real name of Hinduism is Sanatan Dharma, the eternal religion
And your question Why is hinduism, the eternal religion (Sanatan Dharma)???
I differ with your opinion ., hinduism is not a religion , it is only a dharma ., an eternal way to be performed on regular basis ,
nowhere in bhagavatgita Hinduism was mentioned as a religion
Sanatana means that which cannot be destroyed by fire, water, air, weapons and is present in all living and non living being. The nearest apt English word for Sanatana is ‘Eternal.’ And dharma is duty, responsibility of an individual or code of life or law. so, Sanathana Dharma means the ‘Eternal Law.
Sanathana dharma is based on four fold goal
Dharma following dharma, the natural law ,
Artha., money earned in a right way
Kaama desire in the right way /approved way
Moksha unperturbed joy and peace
The concept and practice of social justices is very much important part of Sanatana Dharma
Sanatana Dharma is by its very essence a term that is devoid of sectarian leanings or ideological divisions. This is evident by the very term itself. The two words, "Sanatana Dharma", come from the ancient Sanskrit language. "Sanatana" is a Sanskrit word that denotes that which which is Anadi (beginningless), Anantha (endless) and does not cease to be, that which is eternal and everlasting. With its rich connotations, Dharma is not translatable to any other language. Dharma is from dhri, meaning to hold together, to sustain. Its approximate meaning is "Natural Law," or those principles of reality which are inherent in the very nature and design of the universe. Thus the term Sanatana Dharma can be roughly translated to mean "the natural, ancient and eternal way."
The Sanatana Dharma through its Veda and Vedic literature empower us to aspire, seek and achieve the highest level of knowledge. It provides not only the vision and mission but also the structured approach through its various paths.