"Why?" is given as a backup against a global catastrophe on earth.

Myself I doubt this, there's very few which would affect all of and only Earth. Gamma ray burst? Also Mars. Paperclip optimiser? Probably also Mars. Pandemic? If you're moving stuff between planets, probably also Mars.

Nuclear war and climate change combined won't leave this planet anything like as uninhabitable as Mars, though of course the converse is that the capacity to colonise Mars means the middle of the Sahara, the middle of Antarctica, all superfund cleanup sites, and the top of Mount Everest, all become easy to turn into friendly and pleasant cities you'd be happy to live in.

I hear this and understand the human need to think this is logical. I don't see this as a reasonably back-up plan. The chances of a catastrophe on Mars is even greater considering the environment is not fit for humans.

The amount of ingenuity and co-operation needed to go to Mars may be greater than the amount needed to maintain Earth.

I don't see on Mars is safer than just an orbiting space station. Needing to go to Mars or anywhere other than Earth adds complexity to the issue.

Getting a million people to Mars, assuming Musk's price optimism is accurate, is somewhere around 10% of the upgrade costs Earth's electricity grids already need, or about half the material (not land, not planning, not political) cost of building a new circumglobal 1Ω TW-scale power grid from aluminium.

I would (if I had a say in the matter) go for the Moon over Mars or an orbital space station. An orbital space station (or collection of them) for a million people would be very hard, because we don't have a convenient small asteroid to mine for building materials and therefore can't ISRU. I'd pick the Moon because it's close enough that when the inevitable black swan catastrophe happens, it's only a few days travel so we'd have a chance of saving almost everyone's lives if all the water or food was lost to depressurisation/the outside environment/surprise mould.

Mars, to rephrase what I wrote before, is less hospitable than all the worst environmental disasters we can experience on Earth, even if those disasters were combined. Being able to colonise it means we can fix basically anything that goes wrong here.

Do you think there will ever be a point in the future where humanity will colonize the solar system?

Probably not? Besides potential mining and resource extraction whatever we do to Earth it’s still almost guaranteed to remain more habitable than any other place in the solar system. Also exponential population growth seems not a real threat anymore so it’s not likely we can ever run out of space.