Why in Britain do we stop for a 'coffee', but a 'cup of tea'?

In polite company in Britain one asks ones guest if they have time for a coffee - usually if it is morning. But if it is afternoon one would ask them if they would like a cup of tea.

Now this is not about why we drink coffee in the morning and tea in the afternoon, but why it is that we elide cup of when speaking of coffee, but that it is essentially included when speaking of tea.

Solution 1:

Because in British usage, tea can also mean a light meal taken in the afternoon.

By taking a cup of tea, we are differentiating the action from taking tea. We are not offering to provide a meal.

There is no meal called coffee, so a cup of coffee can be considered to be redundant.