is it necessary for Artificial NN to be fully connected or only fully connected NN is called ANN?
No, the word artificial in this context is only used to separate the neural networks we build from the natural/biological neural networks that they are loosely based on. But in the realm of data science you can basically use NN and ANN interchangeably and fully connected layers are irrelevant to the terms.

Here is a hierarchy that might make things clearer:

Neural Network
 ├── Natural Neural Network
 |    ├── Brain                         (Not fully connected)
 |    └── ... 
 └── Artificial Neural Network
      ├── Dense Neural Network          (Fully connected)
      ├── Convolutional Neural Network  (Not fully connected)
      ├── Recurrent Neural Network      (Not fully connected)
      └── ...