Bitcoin is a proper noun and refers to the protocol(technology) that was created by Satoshi in 2009, developed and maintained by developers from different parts of the world with reference client as Bitcoin Core. Its a protocol for network that can be used to settle payments with BTC or bitcoin as the currency.
There are a few other implementations of the Bitcoin protocol that work in consensus with Bitcoin Core: BTCD, Knots, bcoin, libbitcoin etc.
Satoshi referred bitcoins(lowercase b) for plural. However, people use 'bitcoin' plural as well because there is no authority deciding grammar for it.
If proper, which clients can claim it?
Any Bitcoin implementation that works in consensus with Bitcoin Core as its used by 98% nodes or alternative implementation with different consensus rules that everyone agrees upon could be considered a part of Bitcoin p2p network. This would include users, developers, exchanges, merchants, wallets, lightning community and other economic nodes.
Can a marketing campaign hijack the name? How to detect if it happened in the past?
Maybe. It never happened in my opinion because some early developers are still active and positive about Bitcoin development.