Apart from the most straight-forward one, reducing the risk of losing it, is there any advantage of not carrying both passports with you in case of dual citizenship? Similarly, is there any advantage of not revealing that you have a second passport when traveling, apart from the extreme case of people from a certain nationality not being allowed to enter the country of destination?
Dual national here, and a holder of multiple passports from the same country before I became one.
I usually travel with all the passports I expect to need on a given trip, and no more. This always means my 'home' passport, plus my other passport if there's a good reason for it: returning to 'other home', one nationality is visa-free but the other isn't, etc.
I've never had a problem with carrying multiple passports; quite the contrary, it's saved my ass a few times. For example, once I applied and received a Vietnamese visa in a passport which (I realized a bit too late) had less than 6 months validity left. At the airport, the airline was about to deny me boarding, but I pulled out my second passport, which had more validity left, and they let me board. Showed both on arrival, which earned me a literal raised eyebrow but no comment beyond that, and was stamped in without a problem.
Aside from your "extreme case", though, there are still some cases where multiple passports can land you in hot water. The "extremest" is when one of your countries doesn't allow dual citizenship, and having them find a second passport can lead to harsh consequences, up and including losing your citizenship. (Malaysia and China are somewhat notorious for this.)
Less extreme but still a potential pain in the patootie is the reason I had two passports back in the day: one passport had stamps from Israel, so I had to use the other to enter Saudi Arabia. I still brought both along, since certain other visas of importance were only in the one contaminated by contact with the Zionist entity, but I made sure to tuck it away in a near-invisible pocket of my rollaboard. Again, though, I never had any problems in practice, the Saudis don't really care about anything that's not bottle-shaped when X-rayed.
I would never volunteer having a second passport/citizenship unless that was relevant though. While it may not do you any harm, it's certainly not going to do you any good either, and in less enlightened parts of the world having multiple passports is still considered somewhat suspicious.
The advantage highly depends on your type of traveling and what passports you have.
- If you have dual citizenship of 2 nations in conflict (Israel / Iran etc) I would never carry both. The risk of discovery and the ensuing questions is just too high.
- If you are more likely to lose some of your belongings but not all of them (business travel), take both with you and store them in different places. If you are more likely to lose all you have (backpacking) keep one at home so it can be fedexed to you in case of emergency.
As long as you do not have any huge risk of losing both passports at the same time or being caught having both, carrying both is surely practical.