Here everything depends on what country you are going to go to. If in Egypt, then it is possible and will "carry". It is necessary to understand that the data on the child, pasted on the parent's passport, confirm in principle the citizenship of the child, not his personality.

Therefore, my advice to you: rather make out the passport for the child and travel calmly. And employees of travel agencies claiming that you need a separate passport are partly right, because if you happen to have any kind of incident, then at least it's a moral responsibility to provide reliable information.

I wish you a pleasant journey.

In any case, you will not fly out of the site or from the Internet. And you have a passport at the airport, and not at the agency. Most likely you have a passport of the old type, where children were also entered. Until the end of the passport, it is valid for you and the child for any journey. But, that you would definitely be confident, I advise you to go to the official website of the Russian Border Service.