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i used to work in this specific area and there isn't enough in the photos to be 90% certain that the tree is a problem.
it's certainly possible that there may be some superficial root damage to the structure - but is the structure just a few breeze blocks and some cement over hardcore? in which case it's going to fall down eventually in any case.
not wanting to start an argument, just that there are a number of things to consider and some bloke rocking up saying "your tree is damaging my x" doesn't mean anything - they all say that as an opening gambit as it's relatively easy just to get rid of the tree, then they don't have clear out their gutters, wash their car or whatever the f*ck happens to be annoying them at that moment in time.
My priority would be finding a way to keep the tree.
That doesn’t change the reality though; I’m simply warming @Pantsmaster up to the fact that his neighbour / the garage tenant likely is correct.