Adobe RGB image in a monitor that only displays sRGB?
I'm not sure if here lies a misconception.
A monitor does not only displays sRGB or Adobe 1998, they display a percentage of them. If a monitor displays 100% sRGB color space it will display a percentage of the other (70%-80% ish. I'm not sure at the moment). I am sure you have seen the typical graph using two triangles on top of each other.
The point is what does it mean?
I will try to simulate the differences.
Here is a typical example vendors use on a website to "wow" you.
"Here is a comparison between our (Put name here) and another using standard technology (let's say sRGB)"
And you can think: "Oh, I see that B has vivid colors, and A has dull ones. Take my money, I want B"

Well, probably B has a smaller color space.
What it means is that you reach the peak of saturation faster, therefore you see images more vivid faster. A higher Color Space means that where the other has peaked, you are still on your way.
Imagine that you have a photo with vivid colors (I should use a green or cyan photo, but this one illustrates more the point I want to make.

On a monitor with smaller color space, you probably stop seeing differences in data (detail) let's say between the red lines. But in a wider color space, you can see details further. Imagine the green lines.
(In a real-life, having two monitors present, yes, a higher color space probably will look more vivid, because additional factors, like the overall power of the light, the contrast, etc.)
The peak of saturation?
It is easier to understand with normal contrast.

I obviously cheated on the previous image, I grayed it.
Here I am not. I have pure white and pure black on both sides. But on the left, a more contrasted one (brightness contrast), I peaked the black, and from one point I have only black (orange circle) but on the other side, I have smaller zones of black because I still have differences. The same as white.
This is the same with saturation. There is a point where I can not see differences in the saturation of the colors, mainly on greens and cyan.
would it make any difference to the final output?
No. A final file will be the same, a final file. Imagine that you turn off your monitor when exporting. The file will be the same.
The question in reality is:
Am I previewing accurately the final output on my monitor?
The answer depends on a lot of factors.
Are you really calibrating your equipment?
Are you using your files for print or for electronic devices?
Are these outputs really calibrated?
Do you need a lot of detail on all the images?
Does the final viewer care?
What are the real characteristics of the monitor you are using?
I would say. Some fictitious numbers:
- 90% of the population of the planet, they will not care.
- 9% like to think they care.
- .999% rally care.
- .0001% will really know the difference. You and probably your workmates.
What is your target audience?
The only real way to know is actually immersing in the workflow comparing the two monitors, calibrating them, and comparing them to a calibrated print if necessary.
If the output is for the web, you are fine.
Image source: https://pixabay.com/es/photos/antelope-canyon-arenisca-ca%C3%B1%C3%B3n-1128815/