Learn Portuguese (Portugal): Não como carne nem peixe.

Como is a very common word with two different roles. It can mean 'how' or 'like', but it is also the 'I' form of the verb comer (to eat). Because it comes right after não and is followed by food words (carne and peixe), the context makes it clear that it means 'I eat'.

Is não ... nem the standard way to say 'neither ... nor' in Portuguese?

Yes! When you want to say you do not do one thing or another, you use não before the verb and nem between the two options. Literally, it translates to 'I do not eat meat nor fish.' You do not need a separate word for 'neither' before the first noun here.

Why is there no a before carne or o before peixe?

In Portuguese, you often drop the definite articles (o, a, os, as) when talking about a broad category of things in a general way. Since you are saying you do not eat meat or fish in general, rather than referring to a specific piece of meat or fish, no articles are needed.

Could I say Não como carne e não como peixe instead?

You could, and people would understand you perfectly, but using nem is much more natural and concise. Nem is the perfect everyday tool for linking two negative ideas in a single sentence without repeating the verb.