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.