A Schengen country (like France) will issue a Schengen visa valid for all countries although it is best to be from the first country of entry. Once you are within the Schengen zone, you can move around freely as there are no immigration checks; either at roads or airports. As such, a separate Schengen visa for each Schengen country is a nonsense.
The UK is not a Schengen country and cannot issue a Schengen country. Unlike the US, the European Union is not the Federated States of Europe and does not have one immigration bureau/agency for the whole of the EU. In fact, the UK have many immigration rules that are completely differently to that of Schengen countries. The EU is an economic union only.
The UK and Ireland share one common immigration union is separate from the Schengen agreement.