Can you exchange your driving licence from one EU country to another EU country?

Yes, you can, and the good news is that it's just a formality that's free of charge. In fact, it's mandatory to do so after a number of years have passed since you've moved residence to a different country (in the case of the UK, you can use your non-UK driving license for at most 3 years, after which you're supposed to change it to a British one, but I think most countries have 2 year limits) -- I don't think this is strictly enforced in practice, though.

In the UK it actually depends on where you passed your driving test. For example, consider someone from Argentina who passed an Argentine driving test and has an Argentine driving licence. If they went to live in Spain they could exchange the Argentine licence for a Spanish licence. However, if they subsequently went to live in the UK they could NOT exchange the Spanish licence for a UK licence. I guess the same applies to someone who passed their test in Australia, exchanged it for a UK licence and now wants to go to Spain - but I'd need to check Spanish law to confirm.