The only machine I know that that can cut with both such precision and intricacy would be a CNC mill/router. Are there any more conventional machines that could also be used to make a piece like this?
This is almost certainly done via CNC.
It is possible to make something like this purely by manual processes, all you have to do is look up historical wood carving to see what's possible from the hands of a true master carver (e.g. Grinling Gibbons). But in this case 99.99% chance it's CNCed.
Lastly, you would need a really big block of wood to make such a large piece- nothing like I've ever seen at the Home Depot.
It may not be made from a single piece of wood. In fact it would tend to be stronger and more stable if made from a 'lamination' or glue-up from a number of pieces*. I think I can make out a join line in one of the photos so the starting block might have been glued together from 2-3 pieces. If they're producing these in number I'd expect that they're generally, possibly always, using glue-ups as their starting point.
In the past it was much more common to use single large pieces for a number of reasons. The first is simply that there was much more wood available, much of it of high quality, and from this they could select the best pieces possible. The wood then was exclusively air-dried (drying in a kiln tends to make wood more brittle) and it could have been dried for many years (seasoned) which together make for much better wood. Even today in workshops with high standards pieces of wood can be put aside to season in a storeroom for years after they come in, despite being dry enough to work upon arrival.
Is there a formal name for the "big block of wood" that you would need? And where could you find one?
Depending on the part of the world you're in there are traditional names for pieces of wood of specific sizes and lengths, but often today you'd just buy by dimensions. So you might simply say "I need a piece X by X by X."
In the UK you'd generally need to go to a sawmill or timber merchants (sawmill or lumber yard in the US) for larger pieces.
*Not the same sort of thing but the wooden countertop material offered by Ikea for example, its construction is based on the same principle — even if single boards of beech could be found that were the same dimensions they would be far less stable, virtually guaranteed to warp.