Copyright already de facto works that way; you get copyright the moment it is fixed in the relevant medium, there is no publication or public release requirement. Patents have their own complicated things, but part of the definition of a patent is that it is for public release, so "secret patent" is theoretically a contradiction. (As my phrasing suggests I have little long-term confidence in our systems maintaining that truth, but I believe it is true right now.)
There are secret patents in the realm of national defense (think nuclear). The patent doesn't issue until the secrecy order is rescinded and the term begins once issued; IIRC.
> "secret patent" is theoretically a contradiction
There are "submarine patent" strategies though, which seek to postpone revelation of a patent for as long as possible.
Less of a problem these days though, since patent term is now measured from priority date (whereas it used to be measured from issue date).
It's useful if the startup has something it wants to hide in its trademark application (like a product description that it wants to protect) and can justify the added filing fees. But, there are over 1,000 new US trademark filings every day so in most cases, someone would have to be actively searching for it. Also, often the description that accompanies a trademark application is rather broad, so it may be difficult to determine what the application is for without knowing more. Tech companies like Google and Apple are under much more scrutiny.
IANAL, but it seems like if you have to have secrecy for some reason, it's a solid strategy. But chances are good you'd need a local lawyer in whichever country you want to begin your filing in, and need to pay that lawyer to manage all national correspondence. You'd also have WIPO fees and fees in the US on top of the fees in the origin country. So it's really just cost-benefits.