Fish's brains don't have a highly developed limbic system (emotion centre) like higher mammals etc. So that means they cannot feel complex emotions like humans.
However, they wouldn't survive long if they didn't feel emotions like fear and if they didn't feel pain - anyone who says they don't doesn't understand animal behaviour at all. Fear and pain are useful things because they get us (or a fish) out of danger.
Anxiety is also essential to survival - if the fish wasn't anxious about being eaten, it would probably get eaten. OTOH, fear, pain and anxiety are not things a fish should sense all the time and if they do, it tends to adversely affect their immune systems and they get sick.
Fish that care for their young or mate for life will experience attachment, even if not actually affection. They will also become familiar with an owner and recognise them as a good thing, rather than a dangerous thing.
I would think the closest thing to happiness a fish experiences is comfort. In good water, with good food and the plants or caves the fish requires, a fish experiences minimum fear, pain and anxiety and is able to care for their young, mate and live the way they were intended. I'm a bit like that myself