What's with the last name?

I did not know that our society is obsessed with last names or surnames, until recently. It's said, you are known by your company but few mind sets cannot be changed even if people are sent to Mars! It so happened that I have married a guy with a different surname. Thanks to some much needed modernisation in the society that I was not forced to change my surname or other ways getting the surname changed in all the educational documents and passport would have costed a lot of effort and time and hence my identity was left intact. Then, I became mother of a beautiful daughter and whoa!, I couldn't get much examples where surname of mother is preferred in kids' name, in case parents have different surnames.

Let me tell that at least in India, one cannot delay much in selecting name for the new born as making the birth certificate could unnecessarily become tedious. Now, the name cannot be pre-decided as the gender of the baby is not revealed. I started looking for names and finally got settled with Aananya. Now that's the first name. How about last name? To remove any conflict I could have gone with just first name but that's not good in long run. Ok, I am not obsessed with last names and went easily by adding the last name of my husband to her name. I had still not communicated the name for the birth certificate. I will not pretend here and disclose that I had a feeling of discomfort suppressed within me that why is the baby named after dad's surname and the mothers' identity is lost completely?

One day, with no specific intention, I tried googling how common is the name of my daughter. Alas! little did I know that googling could land me into long term problems! I landed a site which predicted the future of babies based on first and last names. First name of the baby predicted a wonderful future but with the addition of last name, it predicted few health related problems.

People who knows me would understand that I just don't take these things seriously. But somehow, this time, mindset of a silly first time mother won over the general thought process. Let me add that my baby is a pre-matured born baby. 

I couldn't stand my decision of the name given to baby and looked for options of changing the first name. I couldn't take much time as birth certificate had to be made. No better first name than Aananya  could be found, of course per my opinion. So, the option was only the last name. Thank goodness that my husband did not express any hard opinions when I expressed my concerns over the name of baby.

I also talked to one of my college mates who too married a guy with a different surname. :)

I asked her if she too felt discomfort while keeping the name of baby and I was happy to know that I am not alone there but she said she would not want to put herself in any adverse situation, so she went with the flow. Well, I was also going with the flow until I fell into the future predicting site and failure to ignore it.
The name got finalized with the last name of mother and the same got printed in the birth certificate.
Then came the step for making her passport.
We live in very strange society. On one side the entire nation and media goes gaga over the fact that Obama's guard of honor was a lady but on the other side nothing much really happens when it comes to practical stuff. I remember seeing the news of guard of honor on every possible social media and big words like women empowerment, nari Shakti (power of a lady) being used repeatedly.
Guess what, while filling the details of baby for passport, I noticed that the "Name of mother" field was marked optional while father's name was mandatory. I can still not find rationale behind that.

My husband is currently outside India and in one of the most developed nations of World. It must have happened that he showed copy of baby's passport to his friend who out of everything else noticed that the surname was of mother and pointed that it's not usual. I agree it's not usual but is it that difficult to accept?

Could someone please help me understand as to why mother's surname cannot be preferred for kid's name? What difference does that make? Why does our society make it look so complicated?
Do I really owe an explanation to everyone on the surname of my own blood?