Is it incorrect to say "give us" when you actually mean "give me"?
Give us a shout later, lad.
Give us a go there.
Give us a chance.
Give us a few of those.
Give us an idea.
No. It might be inappropriate in some contexts - such as a job interview! - but it is not incorrect. This informal usage is common in Britain and Ireland.
A little like the old saying 'There's no such thing as bad weather - only unsuitable clothing', I'm of the view that there's almost no such thing as 'incorrect grammar' for native speakers - only inappropriate contexts. If you say something in a certain way, and most other people around you say it the same way, and kids grow up copying that way of saying it, you can't say that that feature of language is incorrect. It may be considered dialect, slang, or simply non-standard, but it's correct inasmuch as it follows the rules of the context where it's used.
Take the fact that millions of native speakers say 'You was there' or 'You done that' every day of their lives . If every other person in your class or down your local pub says 'You was..', then it's arguably the 'correct' thing for you to say if you don't want to feel alienated or have people take you for a snob.
Likewise, 'Give us..' is entirely appropriate in some contexts, and subtly different from the standard 'Give me..'
Compare 'Give us a kiss' (which is cheeky, fun, and harmlessly flirtatious) to the more serious and straightforward 'Give me...'.
Or say your mate has some chips and you fancy one. Isn't 'Give us a chip' friendlier than the standard 'Give me'?
An American perspective:
The royal 'we' has a demeaning ring in American English. No one likes it save children who don't yet know that it sounds like they're being addressed as if they're simpletons. The same goes for the insulting 'nurse's we.' Like Mikkel, I've come close to walking out on appointments because of it.
'Give us' to mean one person? I've never heard it used that way. What's wrong with being direct and politely asking your 'lad' (that's a good dog name here) for a French fry? In short, 'me" means me and 'us' has the usual and common meaning of plural.