Unlikely. You'll a) need to check the wheel and tyre sizes and specs of the specified spares for both are the same, b) that the stud arrangement is the same to fit the hub of the Mazda, and c) depending on which Mazda6 it is (gen-1, 2 or current 3rd gen model) whether the underboot area just takes the space-saver only (gen-1 and 2 models) or using a special polystyrene cutout around it that also holds (say) the wheel brace/jack etc.

There's a reason why Mazda charge £395 for the spare wheel/tyre 'kit' - unlike my gen-1 Mazda3, it doesn't just have a standard unit that fits into a nice round hole under the main boot area. Stupid idea, IMHO, as it puts people off buying the gen-3 (SkyActiv) cars who feel they shouldn't have to shell out more for one wheel/tyre that can't do more than 50mph than 2 or even 4 of the other tyres. I won't.

If it's a gen-1 or 2 car, then you could be lucky given your current car is of a similar size and may have a similar, more standard spare wheel and tyre (as many cars of the 90s and 2000s had). Otherwise you could pick up a decent space saver spare wheel and tyre online for a reasonable price - new or 'used' (more likely from a car that has been scrapped after an accident etc but the spare is undamaged and unused). Just be careful about buying on ebay & co - some less scrupulous sellers are on such websites and could offer dodgy kit or very old/used tyres.

My current Lexus IS220D is being scrapped and I am getting a new Mazda 6. The Mazda does not come with a space saver wheel, but my old Lexus does. Can I simply keep the space saver from the Lexus and use it in the Mazda or will I need a specific Mazda space saver wheel?

PCD is the same but the Toyota has a centre bore of 60.1 & the Mazda 67.1. If it was the other way around you could have done something for reuse (assuming same rolling radius) but not this way.