It looks like it's a VPS.
This isn't the same as a shared webhost - it's much closer to a separate machine (except somewhat cheaper). I presume it's a temporary measure until the income is enough to afford full hosting.
It's pretty secure, unless the hosting company is dishonest.. and I expect we'd hear about that pretty quick.
Shared hosting is when usually 100s of websites are stored within one server, that server being maintained by the website host.
VPS is a server that is split up 2 or more times to create virtual servers, kind of like how VMWARE works to give you a good idea.
Having your own dedicated server doesn't at all mean more security, it can infact mean more security issues because they usually maintain their own server. At least with shared hosting, security of the server isn't your problem, but you also have little control.