The US has a homelessness rate of 6.2%, while the UK has a homelessness rate of 7.7%.

That's actually nonsense for the UK, as the real figure is 280,000 which includes people in temporary accommodation, which is provided by the government.

In the UK if someone is on the street and needs a place to stay there are shelters in every town. Nationally there are about 4,000 people that sleep rough, although most of the people that do suffer from addiction or mental illness. Many choose not to go to shelters because you have to be sober to stay in them.

That is astonishingly high and may be counted differently (people living insecurely on couches)

Not sure about the UK, but any kind of insecure or temporary housing is counted for the US - including people living on couches.

Using what metric?

To use a local(ish) example to me, Multnomah County in Oregon has about 815K people, and 3800 that are either living on the streets, in a shelter, or in temporary housing. That's 0.4%. Portland is known for having a homeless problem.

One in around fourteen? I don't need to look anything up to know that's way off, in both countries. I doubt that can even count as 'misleading'.