All IP addresses are governed by regional internet registries (RIRs), in Europe it is called RIPE. There's ARIN, APNIC and others too.
RIRs are responsible for issuing IP blocks to organisations, and also AS (Autonomous System) numbers as well. This is a public database, and you can see all of BT's IP blocks here
https://bgp.he.net/AS2856#_prefixes for example. This shows RIPE has issued BT with AS# 2856 and all the IP blocks underneath it. Central routers use something called Border Gateway Protocol (BGP), which is a protocol that allows routers to communicate with eachother and update and 'broadcast' routing tables. This is what happens in places like LINX and usually any end point of any network when it meets another one. It's not broadcast to everyone, just those the ISP 'peers' with (ie an approved neighbourly network). And so as your packet hops along the internet it gets transferred between these routers all talking to each other and thus a connection is made. It's how the Internet works.
That data is all gathered by companies like MaxMind (https://www.maxmind.com/en/geoip2-databases), which publish both free and paid databases gathering all this information, plus other sources to be able to tell more information, location being the most valuable, on an IP address. The company you're engaging with is likely using this (or a similar) database.