In parshat Miketz, the example of the lack of food is shown in the poor harvest of grain (the common translation of "corn" doesn't refer to the New World grain corn (maize) but rather any grain, or sometimes specific local grains).
Grain is the staple food for people in those times.
If grain is all that is missing, sure, you can eat other things. But why is the grain not growing or ripening in expected quantities?
Most of Egypt gets very little rain anyway. But rain is important further south, to fill the lakes and tributaries that feed the Nile. Crops grew in Egypt only in or very near places where the Nile flooded each year. They had an elaborate system of canals and irrigation to provide water to crops during the growing season.
In the Levant, the drought had a more direct effect, since places like Israel get more rain in the wintertime. In all places, drought reduced the water available for irrigation.
Drought will reduce available quantities of other plant foods. Even established trees (for nuts or fruit) that do not normally need supplemental water will dry up and die (or live but reduce their yield). Other plants will die or have lower yields as well.
One year of drought doesn't generally make a huge difference, though it can for some crops. But here we had 7 years of drought. Drought can be devastating even if water levels never drop so far that people and animals do not have enough to drink.
Okay, so they can just eat milk and eggs, right?
Well... livestock animals depend on grains too. Most livestock gets fed grains or hay (which is grain with the stocks still attached...if it's just the stalks without the grain, it's called straw, which is not nutritious). And most livestock has some access to pasture. What is grass? it's grain. Some animals eat the grass in a leafy form and some need grass that has created grain seeds for their nutrition (or both). Either way, the drought has ruined the pastures. Poultry can eat bugs but they usually supplement with some seeds or grain and there will be fewer bugs on the pasture if the cows/etc aren't eating the grass and leaving manure.
Birds will not lay eggs and mammals will not produce (much) milk when underfed. These things are for reproduction, not survival, and they require a huge number of calories.
Can they eat the animals?
Sure. And they'll have to, since the people won't be able to keep most of their animals alive, so they'll slaughter some instead. That works out well for better off people who have animals. For a year. Maybe two. But we're talking about 7 years of famine. That's a very long time to keep animals alive and continue to eat them. And it doesn't help people who aren't rich with flocks.
What about the river fish?
A lot of people will depend on those fish. In normal years, the river flows high and strong and is full of fish. People will use the fish as a small part of their total food intake. Most will be grain, vegetables/fruit, dairy/eggs, and some meat.
Now, all of a sudden, people will crowd the river, desperate to get what fish they can find. Very quickly, there won't be many fish left at all. Not to mention, that the drought means the Nile isn't as deep or as wide or as strong.
What is a famine?
A famine is not the same as starvation. In any modern country we have starvation. People who go through dumpsters looking for food, people who go to soup kitchens, and so forth. Famine means everyone is suffering. Maybe not the rich, but the majority of people are affected by whatever calamity has caused there to be a lack of food (drought, war, locusts, etc) and there aren't enough other food stuffs or stored food to make up the difference.
Egypt and the Levant probably could have survived this drought for a single year and had it not turn into famine. Seven years, however, is a disaster. The kind that leads to mass death, huge migrations, and sometimes even wars. Even in modern times.