Boris Johnson’s £350m a week for the NHS lie is really just part of the whole meta-lie about the EU: that it’s separate from the UK.
The EU is a membership institution. The UK is (still) a member state. The EU isn’t “them”, it’s “us” and our partners. While the EU has 28 member states of which the UK is just one, we’re one of the big three (the others being France, Germany) that have huge influence.