I see companies which are almost 10 years old still being called start-ups, so... Are there any metrics to differentiate between a start-up and normal company?
I believe that the term has been diluted a lot in the recent years because of hype.
It has to do with what you define as a Startup. One definition, paraphrasing Steve Blank, is that a startup is a "temporary organization in search for a business model", so one you have stabilized your business model, you are no longer a startup. In theory this is irrelevant to time, but if you are searching for a business model for 10 years...
People also confuse start-ups (introducing something new) with companies that start-up, like use a traditional bakery as an example. This happens a lot recently that people are more interested in entrepreneurship and also because there is some overlap. In that case I would say that a traditional business leaves the starting-up phase once things have stabilized/matured (establish a presence, clientele etc), let's say when the rate of change drops. Again 10 years are a lot, maybe not for space/pharmaceutical.
In any case in order to have a "right" answer, we need to look for how people define "startup".
Counter example: try to market your company a startup in order to attract specific talent, while it is not.
Additional material: A taxonomy of startups, from Blank mentioned above: Steve Blank: The 6 Types of Startups.
This is straight from Wikipedia
A startup company or startup or start-up is an entrepreneurial venture or a new business in the form of a company, a partnership or temporary organization designed to search for a repeatable and scalable business model
But in reality I don't think there's a real definition as such, and this one is self-sourced by whoever put it up.
Some countries give benefits to start-up companies. Of course these don't last forever. Some countries can have them last for up to 3 or 5 years. But it's very unlikely that these benefits would ever last for a full 10 years.
I remember my brother having set up his company and having his benefits last for around 3 years. Benefits could include things like reduces taxes for company expenses.
In the end the definition of a startup company will vary in duration. While people can still call themselves a startup company, it would not always mean it is recognized by their government. But it most certainly can vary.