For most locations, the fibre goes only to a (nearby) cabinet (called FTTC - Fibre To The Cabinet). Around here, they're 3-4 ft high. Ones with the Fibre kit inside have air vents and a sticker marking them as having 240V inside (needed to make the kit terminating the fibre work).
Then there is a set of connecting cables from the fibre cabinet to the existing cabinet and the 'data' connection is linked to your copper / aluminium wire to your home.
Yes, OP that last stretch of cable to your home is the same. When I lived in N Wales I had terrible speeds and if it rained I lost not just broadband but voice calls too, for a few days. It turned out that the insulation on the dropwire (from the gutter to the master socket inside the house) had perished. Once there was a new dropwire, everything improved.
Obviously some homes have the cable underground into the street, depending on how long ago the properties were built. I suspect many from the late 80s will have the cable underground to a street junction box and then underground via plastic pipes to each home.
In some areas, there's another option, called FTTP - Fibre To The Premises. I have not investigated exactly what is done (I know a client has this installed, but have not been there in the day to see what is outside and how they are connected.
Seeline - if you have Fibre, surely they have now installed an extra box inside, a VDSL Modem, and your router/ hub is connected to that, for Infinity. If they didn't visit to install, then you have surely just changed to BT Broadband, not BT Infinity.