Purpose of aural alerts (and unclear engine indicators)
Aural alerts are attention grabbing, and they are typically silenced until it's appropriate to handle the issue (barring e.g. a stall warning), i.e. confusion can still arise if the engine indicators are not clear; "Did it just say number 1 or 2?"
But the aviation industry learns from mistakes. The two examples in the question are grandfathered: the 737 is based on 60s certification, and the BAe Jetstream on early 80s, and both use – by today's standards – unclear engine indicators. This was noted in the final report (p. 150) of the 737 crash for instance:
4.8 The regulatory requirements concerning the certification of new instrument presentations should be amended to include a standardized method of assessing the effectiveness of such displays in transmitting the associated information to flight crew, under normal and abnormal parameter conditions. In addition, line pilots should be used in such evaluations. (Made 30 March 1990) [emphasis mine]
(Newer models of the 737 – NG and Max – that have glass cockpits have clearer engine indicators.)
Now transport planes have better indications, including verbal indications in some. One of the early examples to have such verbal warnings is the MD-11. You can listen to them in this airline training video filmed in a full flight simulator at 5:02.
After an aural alert is silenced, engine-specific warnings in text have been around since cockpits moved to glass cockpits. Fuel shutoff controls and fire handles also have lights to indicate which one to close/pull after verification (example on Boeing 777 shown below).
Failure vs fire
Note that failures, as opposed to fires, do not require immediate action in shutting off fuel or pulling a fire handle (memory items as soon as possible), therefore more demanding warnings can be distracting in this situation. For example, note that an engine failure is a "caution" on the Boeing 777, and note the left/right message:
|FIRE ENG L, R
||Fire is detected in the engine.
|ENG FAIL L, R
||Engine speed is below idle.
— 777 flight manual §§ 6.8 and 6.7 [emphasis mine]
Perhaps of note, an early example to have a failure light is the Boeing 727:
Later in the 727 program, in recognition of the difficulty in sensing engine failure [due to the near centerline placement of the engines], a light was installed on the instrument panel to indicate engine failure when it occurred; since this facilitated earlier recognition, the time from engine failure to V1 was reduced to one second, and the total AFM calculation sequence was then 4.73 seconds.
— Boeing Jet Transport Performance Methods [emphasis mine]