Do road signs showing a country and a city as destinations really exist?

Yes, they do exist. The first situations that come to my mind are:

• On motorways when you are approaching the border and there are not many "big cities" ahead. For example: Sign for France
• When you're close to the border you might not want to cross it inadvertently (for example, until a few months ago, I could drive a 125cc motorbike in Spain but not in France). For example: Sign for France
• When the route is not obvious or you have more than one option you could also see a sign somewhat far away from the border. For example: Sign for Andorra, going through toll or toll-free road

There exists plenty, even if they are way common than in old times.

Scope of road signs is not always the same, but generally they should help you move to a destination and at the same time give you a general sense of your position. And you, as a driver, back in the days, had no GPS, few highways, and was traveling by landmarks.

In this case, for example:

• He is in France, near the border, on a secondary road
• The big sign you are referring at, points him to the Route Nationel 7 and in the direction of Paris. Paris is an important landmark, so when you pass the border and you are not on a major road, you are likely to use it as a reference point to get oriented. It's like the sign is saying "North-North West", but in a human readable form.
• The other sign simply points to Italy. When you are near a border you don't get the list of options, because usually there are none: you can cross the border at a specific point, and all that you can do is to go to Italy. So there is no "Italy - Turin" and "Italy - Naples", because there is only a single gate. If you want to go to Rieti from France your first problem is to go to Italy, so a sign saying "Italy" is more than enough
• Last but not least: there is a small sign on the right, that you didn't notice or just ignored, and that is not appearing in the picture. It's a local sign pointing at the road to -I think- Menton. So, the big sign give you hints on how to get out of the local road and go in the direction of two mayor landmarks, and the small local sign help you to find your way around locally.

And just remember this is the border in France ages before GPS, Schengen, highways (at least in France) and so on. It's not like today, that you can cross the border everywhere, on a major highway connecting two nation, and things like that.

Ok, here we go. Sadly I couldn't find the exact one, but this is the same just in the opposite direction:

This is from the Italian side of the border.

There are several signs here in northern Baja California, Mexico, that reference cities in both Mexico and USA e.g. Tijuana and San Diego, and once in Tijuana most signs will have San Diego and an area or street in Tijuana rather than a city.

But there are definitely some, particularly close to the border, that have both Tijuana and USA.

One example I could find an image of easily was a pedestrian sign

Same on the other side where it may say Chula Vista and Mexico or San Diego and Mexico.