Is there a reason why the "x" is not visible in Firefox with Input (type search) fields? And can this be made visible in Firefox? 

Chrome:

Edge:

Firefox:

Have you tried inpsecting the styles and seeing if there is something to adjust in the css?

Is that 'X' something you built or does it come as default?

Chrome and Edge has an 'X' to clear inputs as default (which you can hide), but firefox has not. 

If you are using the default one, then for firefox you would need to build one yourself

I think that this feature is non-standard in Firefox for the tag type="search".

See: https://developer.mozilla.org/en-US/docs/Web/CSS/::-webkit-search-cancel-button

You can to try create in CSS.