You should be editing in the sRGB document color space, and exporting/saving the images with that color space embedded. Therefore, the color space capability of the monitor beyond sRGB isn't really relevant.
However, a monitor that is "sRGB" does not mean it has only sRGB colors, nor that it has all of them. So a monitor that is capable of more colors, and all of sRGB, may be better.
Editing in a color space with colors that your monitor cannot display (e.g. ProPhoto) means you are making edits you cannot see... you can do that, but you would want to softproof your image in sRGB before saving it as an sRGB jpeg/png... if your monitor is only capable of sRGB (or less than), you are already "softproofing" in effect. And likewise for editing in a larger color space you can see (e.g. Adobe RGB (1998) on an Adobe RGB (1998) capable monitor). You might want to work in a larger color space if you are saving the working document (layered tiff/psd/etc) in order to retain as much original color data as possible; for some possible future use... i.e. advancements in display/print technology.
And then, for your own benefit you want to calibrate both monitors to be as accurate as possible, while looking as similar as possible... tradeoffs will probably have to be made.
But in the end, you have no control as to how the image will be viewed on the web... so all of this concern and effort on your end is largely wasted.