I'm not an expert, but I'll try to answer your questions anyway
You ask if the company uses its own blockchain, or does it use the bitcoin blockchain. Do you mean bitcoin blockchain concepts, code, or algorithms? Or do you mean the actual blockchain that currently runs bitcoin? Personally, I think the answer is no either way.
The actual real-world bitcoin blockchain can't be hijacked for use other than moving bitcoins between wallets, so they're obviously not doing that. Blockchains based on the bitcoin concept and/or code are called alternative blockchains. I don't think this is the case either as bitcoin's blockchain is a public blockchain, which doesn't suit DCC's needs. Take a look at the definitions here. I believe DCC would be using a private blockchain.
Next, you said the issue with using an own blockchain is that it can be quite easily overwritten. I don't think this is necessarily the case. Using bitcoin as an example, the blocks are getting harder and harder because the puzzles are getting progressively harder. You could just as easily make it cryptographically difficult to make a new block on a private blockchain if you wanted time to be the limiting factor of mutation. Or, you could simply require credentials to create a new block. This would just mean that new blocks would need to be signed by a private key (see asymmetric encryption).
Lastly, the owner of that blockchain has access to all private info of the users, i.e. the transactions they made, etc. Isn't that an issue? I don't think this is an issue as it's not really any different to the way banks or other company's storing financial data work currently. You simply need to follow standards, have access controls in place, and have auditing/logging to keep track of what's being looked at.
For reporting and auditing purposes, both internally, for regulators, and for partner financial institutions, they'd need to be able to see all of this anyway. It's not unusual, but maybe I'm missing your point?
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