TL;DR;
EM (or EOM as in early documents - and Unicode as well) was and is widely used in data transmission to mark
- the physical end of a medium,
- the end of the used portion of a medium, or
- the end of meaningful (wanted) data on a medium.
(as Scruss already described in his answer)
It was used with most RJE stations as well as with terminals. The confusion/missing information may be due to its quite low visibility to average (terminal device) programmers, as it's used at the core of many data protocols - especially in finance. Each and every credit card transaction will go over various communication links encoded with EM in certain fields.
The Long Form:
But I am unable to find anything more on the "end of medium" character (EM, 0x19, decimal 25) than that name among listings of all the ASCII characters.
That may be because its most prominent usage was within the (IBM forged) Mainframe world. It denotes the end of a variable-length (or shortened) input or output. On some environments it could be used to shorten transmissions to transmit only the characters entered within this (logical) line, not the whole line or the whole screen (keep in mind, mainframe terminals operate in block mode), saving quite some transmission time.
Unbeknown to most, all remote (terminal) I/O in mainframe systems is (usually) ASCII-based (*1). To avoid possible code issues, EM belongs to the group of characters with the same encoding in ASCII and EBCDIC.
The usage developed over time. Generations of terminals, including the 2260, 2741 and 1050 series, had implied handling (by message framing) of EOM as it was called back then (and still is in Unicode). For example, for explicit display the 2260 used ASCII X'21' or !. Later it became available as a user-handled function.
EOM was officially integrated into ASCII-1967 (USAS X3.4-1967) as EM, mostly to satisfy the needs of interoperable RJE handling. The IBM 2280 Data Transmission Terminal manual (GA27-3005-3 of 1971) describes the EM function in context with input/output media nicely for handling of punch cards:
EM (End of Media)
The EM control character is used to indicate the end
of a record when transmitting records of variable
length. The EM character is punched in the card
column following the last column containing data.
When the EM character is read from a card it prevents
further reading of that card. The transmitting
2780 then generates automatically and transmits an
end-of-record (US or IUS in EBCDIC), or end-of-block
(ETB) character following the EM character.
Handling - or more likely, generating - such input might have been the most common usage for ASCII-based environments when hooking up to the IBM world.
What was this character intended to be used for
Intention implies that character sets were created to form an intended use. In reality it's usually the other way around: there are many usages in many different environments; to form a standard, committees try to squeeze in as many different functions, while finding overlapping usage to reduce code points needed.
(AKA what sense of the word "medium" was meant)?
In its most pure sense of any medium: no matter whether a line on a screen, a (part of a) punch card, a paper tape or magnetic tape or some data stream.
Was it ever actually widely used for that purpose, or any other?
Yes. With any credit card transaction you send several data blocks, and EM will be used in some fields. Mainframe will never die :))
*1 - RJE stations like the 2280 can often operate in EBCDIC or ASCII.
