Spectrum of windowed version of original continuous signal
If you don't understand the difference between the Continuous Time Fourier Transform (CTFT), the Discrete Time Fourier Transform (DTFT) and the Discrete Fourier Transform (DFT), now would be a good time to read about them. The very short version is that the DTFT yields the (continuous-valued) spectrum of a sequence (i.e., a sampled signal). The DFT computation results in a sampled version of the DTFT. To apply the DFT requires a finite number of samples (i.e., a time-domain window) whereas such restrictions are not placed on the DTFT in general. On the other hand, the CTFT deals with continuous time signals. There is a lot more to all of this, and I recommend you read more. 

It is true that a complex-valued signal $x(t) = \exp\left(j \omega_0 t\right)$ and the Dirac delta function form a CTFT pair. I can agree with you that the spectrum of this signal is discrete (nonzero for a finite number of frequencies, in this case a single frequency).

After applying a rectangular window in time to the complex-valued signal $x(t)$, the CTFT is a frequency-translated sinc function centered at $\omega_0$ with a lobe width inversely proportional to the size of the time domain window. This shows us that the result of truncating a time domain signal with infinite support in time and a discrete spectrum in frequency can lead to a new time domain signal with finite support and a continuous spectrum. So, the answer is yes, the interpretation given in the question is true.