In bars, alcohol is served by a specific measure. Sometimes bars have bottles mounted on the wall in a measure bracket so they can put the exact measure straight into the glass. If they are pouring from the bottle they may use a measuring cup. A "single" refers to one measure, and a "double" means two measures in the same glass.
In the UK, measures of spirits such as whiskey or vodka are normally served as 25ml measures, so a double would be 50ml. This may vary in other countries. You can't always visibly recognise a double measure if someone orders a spirit with a mixer or as part of a cocktail - the glass content looks the same but there is twice as much alcohol in the mix.
Wine by the glass (as opposed to when you order the whole bottle) is also measured out. In the UK a standard measure of wine is 125ml (if you ask for a small) or 175ml (large). A "double" would have to be two of these measures. Again, these volumes may vary in other countries. Because wine is not usually mixed, it could be visibly recognised that someone has a double measure in their glass, as appears to be the case in your example.
It is also noteworthy that wine is normally served in a large glass with room to spare to allow the aromas and flavours to emerge. Again, this would make it evident if someone had a larger than usual measure.