It is a bottlebrush indeed - precisely, its Callistemon rigidus, info in the link below

https://www.rhs.org.uk/Plants/2693/Callistemon-rigidus/Details

One thing I will say - I grow Callistemon splendens, but it behaves the same way. You will notice that new leaves appear above where the flower is, which means you end up with a bit of bare stem between the upper and lower leaf growth within a year where the flower and its seed pods were - I always cut off the flowers as soon as they start to fade, including any tiny new leaves above the flower, clipping them just beneath their base, and then new leaf growth arrives a bit later. That way, you have a nice, bushy plant, and no bare patches on the stems...