The phrase is sort of contextual, and many confuse "radom" with "indexed" just to make things a bit more confusing.
For example in the context of many Microsoft Basics prior to VB.Net there is a "random file" type that treats UDTs as "records" and allows seeking by record number. The format of these persisted UDTs varies somewhat from their deserialized in-memory format, and most of the details of that are covered in the manuals. Indded, if you dig into the OLE documentation you'll find that the underlying data structure is called a "record" and is
not really a C-style struct at all! See
IRecordInfo Interface.
However for most other files Microsoft Basics treat them as streams of bytes, for text I/O statements the bytes are considered ANSI or DBCS characters. Usually that means 1 character = 1 byte (except when it means 2) so it is possible to seek but by character/byte instead of by record.
Most of that goes back to the weak filesystems supported in Windows and MS-DOS, that hearken back to the weak filesystems in old minicomputer OSs (Microsoft was crippled from the start by overexposure to frail operating systems, in particular those of the old PDP-10). For most purposes all of these OSs only support one file organization: stream files. That's because they barely supported disk at all, treating it as a collection of simulated punched paper tapes.
Once you leave the impoverished world of Windows, *nix, etc. things are different. Stream file support was fairly rare in the past even though it was often added by the 1990s, primarily to foster data interchange. There the major influence was punched cards rather than paper tapes.
Instead filesystems tend to offer record-oriented file organizations and often made a sharp distinction between sequential-only files and random access files. There is almost always some filesystem-level provision for various ISAM-style keyed/indexed file organizations as well. All of those come in lower overhead fixed length record and higher overhead variable length record file kinds.
The reason mainframe OSs tended to have so many file organizations, often conceptually the same, is that the business was competitive. To get business from IBM customers, Burroughs, Univac, Honeywell, etc. typically implemented IBM (and other) formats in addition to their own. This made it easier to convert a customer to your own products.
The reason you see so much chatter on the Web is that the Web is dominated by the bottom of the pyramid: the most people are at the lowest levels of skills and experience as well as least exposure to powerful filesystems. Since in the shallow end of the pool (PC OSs) any form of random access files must be an application level abstraction laid over the dumb stream file... constant jabbering about it is almost inevitable.