Why do immigration officers ask to see ID other than your passport?

It is most probably a simple double-check in order to confirm you're the actual person that the passport represents. Most probably this is prompted by your recent looks being very different from your picture in your passport.

Yes a passport (with a valid authorization) is sufficient to consider your application for entry, but before that the officer needs to be satisfied that the passport does actually represent you.

Whether you should provide another ID or not is subjective, you decide. For me, I don’t play right or wrong with immigration officers. If they ask for something and I have an answer, I do provide it regardless of whether I should have had to or not.

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Anecdote

When I posted this answer, the OP didn't really seem to believe that this answer was right. As fate would have it I was naive enough to fall victim to this scenario and guess where? Russia!.

My passport has a 3 year old photo of mine without any beard (with mustache) and for the World Cup I'm sporting thick beard which took months to grow. My FAN ID has my picture with the beard. I don't know why it never occurred to me that the mismatch with passport could be a problem, I was going to Russia and I should have known better about Russian Immigration staff.

As soon as it was my turn and I handed over my passport, the Immigration officer started alternating his gaze between my face and my passport. Those were the longest 2 minutes of my life. After he was done trying to match my face with the picture he started scanning my passport to check if it was fake. He checked the bio page under a magnifier with some special light, then checked the stitching on the passport over and over. Then he did the same to the very own FAN ID that Russia issued to me.

After he was done with the verification of documents he asked me:

Do you have a ticket for the match?

Yes, here it is

Do you have a return ticket out of Russia?

Yes, here it is

Do you have any friends with you today?

No

Do you have hotel booking?

Yes, here it is

Do you have any other ID with you?

Yes, here are my 2 previous passports and my driving license.

At that point he started to return the documents one by one to me and at the end stamped me in and said Welcome to Russia, Have a good night.

Guess what? The immigration officer on exit repeated the same process to verify if my passport and FAN ID actually belonged to me. It was my first time in Russia and I Love it, but Russian Immigration is the last authority you want to take such fashion chances with.

Things better match up!

Note: In all fairness to them it was my own fault, and they were just doing their job and at no point I was put under undue stress or taken somewhere other than the counter I reported to and apart from the longest 10 minutes ever, everything went well.

Going to roll some comments from Hanky Panky's answer into another answer:

It's likely because they have some doubt that you really are the person who is shown on the passport. This could be because you don't look like your photo or for other reasons - maybe something completely unrelated to you specifically. Perhaps passport fraud/counterfeiting is high for people coming from your country?

Passport officers are within their rights to ask for further information to satisfy themselves that you are who the passport says you are - they don't just have to say "Yep, saw a passport!", they have to be able to say "I am confident that the person trying to get into the country is the person their passport says they are and I have no reasons to believe that they are dangerous or are likely to try to become an illegal immigrant" etc. The best rule for getting through immigration control quickly is to be co-operative so if they ask for another form of ID then it's best to give them one if you have it. If you do not then they will likely spend some time asking further questions and there is a chance that they will deny you entry if you cannot satisfy them.

Ultimately, for the reasons why they've asked you, all anybody on here can do is speculate. If asked for a second ID again you could say "Yes certainly, but can I ask why I need to do this? I get asked often and I am wondering if there is anything I can do to make this process smoother". Be polite and be co-operative and it won't be a problem. This is the only way you'll get a real answer to your question.

Edit: You changed your question to include:

If not why they don't write on passports "Please carry other IDs when ever you or others see a change in your look (short hairs, long hairs, hair loss, weight gain, beard, heavy makeup, light makeup etc.)"

Well I can't speak for other countries but the UK says this:

https://www.gov.uk/changing-passport-information

You need to get a new passport if you change any of the following: ... your appearance, if you can’t be recognised from your passport photo any more (for example, you’ve had plastic surgery)

You don’t have to tell HM Passport Office or get a new passport if you: ... change your appearance slightly - for example, dye your hair or grow a beard

So really the advice is that if your appearance has changed to the degree that you might not be recognised from your photo, for any reason, then you should get a new passport with updated photo.

Also if you are coming back home then no they could not send you back anywhere but you might be in for longer immigration processing and possibly even detainment if they cannot be satisfied that the passport is really yours.

I would like to answer to the edit:

If not why they don't write on passports "Please carry other IDs when ever you or others see a change in your look (short hairs, long hairs, hair loss, weight gain, beard, heavy makeup, light makeup etc.)"

Because they never write rules on the objects, but on the manuals of those objects: they don't write the rules of Monopoly on the board (but in its manual), like they don't write on banknotes "don't burn this", even if it is illegal to do that (the manual, in this case, is the law).

And finally, they don't write on passports:

A valid passport or a National Identity Card is accepted in most cases

because they write it in the Terms of the travel company, such as Ryanair's Terms for Travel Documentation.

Other examples vary from country to country, and WhatEvil's answer gives another example, valid in UK, while in the comment of the same answer there is a more extreme example, valid in India, that includes in the "change of appearance" even growing a beard or wearing a turban.

According to this, asking for a second ID is legit in most of the cases, and is not a whim of the security, it's a written rule.

I've been asked this at London-Gatwick: the only photo document I had was my national ID card and the officer wanted another document, but unfortunately I had nothing else with a photo, so showed my bank card with my name on it.

I think it's because I don't look Nordic (I'm from Sweden, but also have Hungarian and Armenian descent) and the officer had her doubts about me actually being an EEA citizen.

Another time, whilst exiting Serbia for Hungary, I got to show Serbian border police my university ID (with a photo) as they thought I looked too different on my national ID card.

I actually very recently was questioned at the UK boarder as a passport officer didn't seem satisfied with my passport. He asked if my name has changed in any way or if I ever spelled it differently. Then asked for other form of ID. I had on me an UK university student card and NUS card and these seemed enough. He also asked me where I work and what I do.

I never had this situation before and I'm originally coming from an EU country...however must admit I look like a terrorist in my passport photo :D.

I don't know for sure, but this may be a reason: passports (at least the non biometric ones) don't contain a unique personal id - so if they "don't like you" for some reason, and you try to enter to the country with a new passport, they don't have an easy to know who you are.

There is at least one possible reason for this:

If your passport is modern enough, it should contain a cryptographically signed digital copy of your passport photo, plus some biometric data. So if someone were to tamper with the physical picture inside your passport, it should be easy to detect by comparing the digital photo with the physical one.

Those with a European passport can check this for themselves if they have an Android phone with NFC capabilities. Just use this app (I don't think the app will work with US passports and Chinese passports, but those two types of passports use a similar mechanism as well -- even if they can't be read by that particular app).

For African passports, it's a little different. I just googled it. And some African countries have that kind of passport, but many do not, so that older type of passport should be easier to tamper with.