In the case of Bitcoin, it's not clear yet who it's going to keep the name at the next fork. You can see Coinbase blog post about this https://blog.coinbase.com/clarification-on-the-upcoming-segw...
Brands are words that need to be trademarked, but "a cow" does not need to be, because everyone knows what a cow is, and if I sell you what I say is a cow but is actually a cat and then explain that that's now the established brand - you'd say it's cat, not a cow.
The test for Bitcoin is very simple - fire up an old version of the Bitcoin software, and if it validates the blockchain, then the blockchain you have is Bitcoin. If it doesn't, it's not Bitcoin.
And if/when Bitcoin Core ever changes the software such that it is a hard, backwards incompatible fork, then on that day they should stop referring to it as Bitcoin, because it is not the Bitcoin that Satoshi originally started.
If you mean starting an older version of Bitcoin Core after the on-disk storage has been converted by a newer version, then yes, of course. The way data is stored has changed, not the least of changes being Leveldb instead of Berkleydb.
But at the network level newer nodes will communicate with older pre-Segwit nodes just fine.