Yes, air is matter — because it satisfies both tests.
- Air occupies space. Push an inverted empty glass into a bucket of water: the water cannot rise inside, because air is already occupying that space. A blown-up balloon or a pumped cycle tube gets bigger for the same reason.
- Air has mass. Weigh a deflated football and weigh it again after pumping it hard — the second reading is a little more. The extra mass is the air you pumped in.
We cannot see air, but seeing is not one of the tests. Anything that occupies space and has mass is matter, whether or not our eyes can catch it.