Is it possible that the potatoes cross-pollinated with tomatoes?

No, Tomatoes, eggplants, peppers and potatoes are in the Solanaceae or nightshade family, which are self-pollinated.

Under favorable weather conditions, potatoes produce fruits. The green fruit that is observed is really the true fruit of the potato that contains many small seeds. The fruit, although it looks like a small tomato, is poisonous and must not be eaten.

Although they have seed, potato fruits should not be saved for seed because progeny does not come true.

The best way to handle them is to remove these fruit once you see them. That way the plant is not putting energy into produce the seeds and can focus on making better potatoes.

Removing them right away also removes the temptation of children or trainee gardeners to eat the poisonous fruit.