Is it possible to physically "feel" the growth of the hair transplants after 5 months?

You can't "feel" hairs growing.  If you are feeling something, then it probably is something else.  It may be a skin or scalp issue.  It is always a good idea to follow up with your doctor for an exam.

What you are experiencing might be folliculitis or inflammation of hiar follicles that are not uncommon in the first few months after the procedure.   It is good if you can visit your doctor and get it checked by him.  Examination of your skin is the best way for you to make sure everything is okay.

at this point some of your hair should be breaking through the skin to come out. Some of our clients describe itching and tightness when this is happening and may be that is what you are experiencing.

The feeling you are describing may be the result of hair breaking through the skin, but in the 23 years of doing this type of surgery, I rarely heard that from my patients. Most men at 5-6 months will feel the actual new hair if they touch it and certainly see some of it by then.