Is each block found just a link in the same long chain, or are there multiple chains?
The Bitcoin blockchain converges to a single-file chain because each block references exactly one predecessor, and thus there can only be one block at every height in the chain connecting the genesis block and the chain-tip.
However, at times more than one block will be found at the same height. This happens either due to the mere happenstance of two miners discovering a valid block in the same instant, which is benign and will rectify by itself when one of the two sires a successor block and thus creates a new longest valid chain, or it could happen if a faction in Bitcoin adopts an incompatible protocol rule change (hardfork). Since network participants that don't accept the rule change will not accept these blocks as valid, two chain-tips may persist henceforth.
Recently, the hardfork plans of Bitcoin Unlimited have been a major topic in Bitcoin, which frequently includes discussion of possible scenarios with two incompatible chain-tips. I assume that you seeing a discussion of that sort gave you the idea that there could be more than one chain.
Every time Bitcoin is forked, the blockchain is forked too, thus creating a different blockchain. This applies to all cryptocurrencies that were forked from bitcoin.
Even the main Bitcoin branch has at least two different blockchains: one has real value because it's expensive to mine bitcoin there, the other is the test blockchain (testnet), where bitcoins are easily created, thus they have no value.