Is there a web browser or GMail plugin that would allow me to select the text shown in an image attachment?
For example, if I get a screenshot that has an ID or something that I need to copy it would be nice to be able to select it.
No, because images are not the same as text. Text can be selected and can also be rendered with different fonts. Images, which are typically an organized grid of pixels (or sets of formulas in the case of "vector graphics"), that contain what appears to be text are just drawn (like art), often from what we call "rendered text."
You may be able to use some OCR (Optical Character Recognition) software to interpret the text present in an image and provide you with a textual representation of it though (if folks have suggestions for good OCR software for this, please feel free to add links to comments in response to this answer).
It is highly unlikely that a browser plugin will itself handle any OCR. What is more likely is there are full-blown OCR applications that have plugins that enable integration with the browser.
In a brief web search for "OCR browser plugin" TiffSurfer was a prominent result: http://www.visionshape.com/tiffsurfer.htm
Also, while I tend to not like sites like this, it may have apps worth checking into: http://windows.podnova.com/trends/ocr_plugin.html
I cannot personally vouch for any of these applications as the only stand-alone OCR app I ever used was Caere Omnipage and that was back in 1998. It saved me a lot of work when my brother caused a catastrophic computer crash one week before the end of my first semester of college, before I was familiar with data recovery techniques. :)