Does the shape of a wire affect the electric current in it in any way?

Yes, there is a difference.

If you made the wire as you mentioned, into a spiral, like this:

then there is quite a big difference between this and a straight wire. The difference between a straight wire and a coil or spiral wire is that the spiral wire resists changes in current flow.

This is called an inductor or solenoid.

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It resists changes in the current flow because when there is a changing current in the coil of wire, there is a back EMF that counters the EMF of the current entering the coil of wire. This makes it hard for current to pass (at least initially) through the coil of wire.

Inductors are characterized by the following equation:

$$v = L frac{dI}{dt}$$

Where $v$ is the voltage, $L$ is the inductance and $frac{dI}{dt}$ is the change in current with respect to time or how does current change when time ticks.

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When the current is unchanging, the inductor behaves just like a normal wire. But when the current is changing, it behaves differently: the greater the change of the current, the greater the effect of the back EMF will be. At very high frequencies, the current will be mostly blocked, and theoretically, as the frequency approaches infinity, the resistance or impedance of the inductor will be infinite, and no current will pass.

Although straight wires also have inductance, it can be usually ignored because it is so small. Only in certain situations such as power line transmission, it becomes important to know the inductance of the straight wire where it cannot be ignored from the math when designing the power lines.