That's incorrect and somewhat misleading. It's rather such as - certain parts of (minor subset) some civil laws or family laws come under respective personal laws based upon religion/tribe &c. Like Hindu Marriage Act, Hindi Code Bills, various succession acts &c.
Remaining (most) part is a uniform civil code. However, it can be noted that in case of criminal matters - AFAIK - Indian constitution doesn't discriminate, in any manner, at least on paper.
Politics - both religious and generic - is embedded to everything in India, even laws. There's a landmark Shah Bano case where Supreme Court of India overruled the verdict given by Muslims' holy book which was as interpreted by clerics which is often misleading as in any religions. The interesting part here was that the court was compelled to quote from their Holy book Quran in the verdict - an aayat (like some sort of support) - as it was a very sensitive matter.
There are clearly different laws that apply to different religious groups when it comes to polygamy and other, as you say, family or civil laws.
I wasn't trying to say that the entire legal system in India is bifurcated (multifurcated? =P) along religious lines. I was just pointing out that while the American legal system attempts to exclude religion from the law (to varying degrees of success), Indian tradition includes laws which take into account various religious traditions and customs.
It's just a different way of treating the issue of freedom of religion in the legal system.
I just wanted to clarify the scenarios where laws are intertwined with religion, tribes or caste. Regarding the last term caste I have not been able to find any instance but have read in articles that there are such provisions.
I would like to add that your mention of Polygamy in India needs few lines - it's not enforced in India until and unless you are someone famous where people take interest out of the celebrity nature and hence state intervenes or the case/issue has blown up. It's present even in religions other than Islam. It's 'sort of' legal in most of the tribes based on tradition (this term often comes up in all sorts of legal cases in this country and is given importance) and it's common in rural areas.