Why can't an electrochemical cell be a single cell?
Yes, if you dip two different metals in an aqueous solution containing ions, you will have an electrochemical cell. This is how all lemon batteries work, which consist of zinc or copper inserted into lemons (approximately an aqueous citric acid solution). You can see a potential difference between zinc and copper. The key point is that one cannot determine Nernstian potential in such cases.
Half cell is just a mental construct as it cannot exist alone. If there is a reduction, something must be oxidized. Electrochemists usually separate the "half-cells" in order to conveniently calculate Nernst potential.

Imagine a hypothetical cell, which is partitioned into half cells with a plantinum sheet. Each compartment contains an electrode Cu and Zn, dipped in their respective salt solutions. What will happen in that case?