Hello I am from Australia and the term " hug stuff" means to put your arms around objects. This term is mainly used to describe how someone feels a great caring emotion towards things.

'stuff' refers to something or someperson (or both) "Everybody hugs something or someone". Sometimes you may see/hear 'stuff' dropped - "Everybody hugs"

"Stuff" is used colloquially to mean "things taken together". So I have stuff in my kitchen that I don;t have space for, I have stuff I need to do today, there is stuff everywhere in this house, etc. It doesn't mean living things, but here it is used with irony and humour. So it's a humorous way of saying "we hug people/things"