Can you generate 230v alternating current by having a 230v motor rotate its shaft at the same speed?

Answer

You can indeed do this.

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The machine you need for this is a ‘synchronous’ AC machine. It can work both as a motor and as a generator. If you drive the shaft of the machine (for example with a speed of 50 revolutions per second, which is our mains frequency) you generate alternating voltage. This principle is used in all power plants.

A DC machine can also work both as a motor and as a generator. This produces DC voltage in generator operation.