I don't know about other companies, but depending on the type of startup you're running (eg. a SaaS vs. a free service with ads), some things that could be focused on are:
Month-over-month growth in:
User numbers
Sales
Length of time spent on site/app
You could also possibly measure year-over-year growth. As far as comparison goes, perhaps using a stock-market tech index will be a good way to determine if you are experiencing good growth (although this may not matter if you are not targeting massive growth and an IPO/acquisition).
The problem I personally see is that as entrepreneurs, we/them will tend to focus on measuring every metric (through some template we found on the internet), yet it makes more sense to determine what metrics truly matter (determining what is important is tough for somebody not experienced in finance/accounting).
"What are you doing today to measure them?"
Probably using analytics tools (a big part of the SaaS market I think) and also tracking your finances through finance tools (another part of SaaS).
Oh, and per your title question, I think all companies bother with forecasts. They give a sense of direction, in terms of what your growth trajectory is (you can also use forecasts against actual numbers to see why you couldn't mean targets and using that info, spot flaws/weaknesses in your business).