Barbershop isn't a scent, it's an idea. One that is interpreted into a scent by the maker. A barbershop scent is one that evokes a memory of walking into the barbershops of our youth. But no two barbershops were the same, so that idea can vary greatly. For some the scent would be a medicinal, clean, and Barbacide kind of smell. For others it might be a pipe tobacco and cedarwood smell. Others would associate it with the aftershave that the barber used on everyone, which was very commonly a Bay Rum or Clubman type scent. Some makers would take bits of all these notes and blend them together to make their version of Barbershop.
There are some barbershop scents that I like, but most I do not. They do not speak to me, and give me a memory of the barbershop. There are so many scents that some think of as barbershop, that I do not. Proraso Green? I've never thought of that as a Barbershop scent..... GTB, ugh, no way. I know, I know, Barber is in the name.... but it smells nothing like babershop to me, and evokes no good memories. To my nose it's just bad.
For me, it needs to smell like Jeris Hair Tonic. That's the ultimate barbershop scent to me. Bay Rum is also a good one..... But really, just about anything could be considered a barbershop scent. It's a really loose idea and category. Just go look up top babershop scents, or list of barbershop scents...... many that I'd never consider babershop. (Pinaud Lime Sec..... for example was on a list I saw).