OK I've reread your post and have had a few more thoughts!
my CSS font-size declaration:
font-size: calc( 0.2em + ( ( 0.672vw + 0.672vh ) * 1.3 ) );
...I use on both the main website/page and any iframes.
Thinking about Minnit Chat and on the desktop for just a second, how is it used? In an iframe? Clients use various iframe sizes? Full screen?
I would say probably rarely full screen and if so in a popup (so not full screen so much as full browser width in that case) so probably kept to a small window so the user can do other things on their desktop.
In an iframe it's probably mostly used in a largish iframe and rarely a small column size box. Here in my test site it's used in a 'medium' size I would say :-) Something in between a column and full width of a fixed width website design.
So if you have Minnit Chat there in front of you and at the size you would normally use it, after adding that font-size declaration, you would probably have to now alter slightly a few other font sizes across the Chat window until they are the sizes they were before (so it looks like your current default css).
I haven't looked at the Minnit Chat css but that makes me think perhaps there are mixed em and px units or something as I would have expected those 2 sizes of text to have scaled equally.
If you quickly reset a couple of font sizes so it looks right, then you will be able to see the effect of the whole chat scaling including font size, as you scale the browser window (or alter the size of the iframe if you are testing with it embedded).
At this point I would be testing on a computer only, and without any initial scale tag. After adding the initial scale tag, on a desktop browser it should look identical (from what I have read about initial scale).
So effectively it should not vary wildly any more. Your main issue!
In my test page here, If I size the window to full screen then it does not look comically large! I am on a 13" laptop screen as I write this, and I would never have my browser windows full screen. About 80% of the width at most.
Thinking about users of far larger screens on the desktop, whilst they might make a browser window full screen, I don't think they expect the readable part of the webpage itself to be full screen, so testing a Minnit Chat full screen on a huge screen probably would look comically large yes, but I'm sure not web designer embedding Minnit Chat or user would intentially try to use the Chat in such an extreme size!
Now a whole separate issue is the viewport initial scale tag. If all mobile browsers that support it, support it in the same way as there doesn't appear to be an official spec, and if it's necessary in the Minnit Chat code.
If you have time, please reset the 2 text sizes and test a bit to see how it scales with various browser/frame sizes! It might be interesting to see. Another issue is the heights of the various boxes also not being scaled in view units or percentages, and they could perhaps be if the font scales!