Is it an affair if you are already separated?

In some states, yes.

https://info.legalzoom.com/can-married-men-legally-separated...

> In some states, adultery is a crime, although it is rarely prosecuted. In states that still recognize fault-based divorces, adultery has more of an impact. If a man begins dating during a legal separation in one of these states, and if his wife can prove that the relationship is sexual in nature, she can usually file for divorce on grounds of adultery. This can affect issues of property distribution and alimony. However, some states, such as North Carolina, make a legal distinction between dating during separation and dating while living together as man and wife.

“Affair” is not a legal term in any US state so far as I’m aware.

Adultery is, though. Which an affair is likely to include, and this one apparently did.

Bezos lives in Washington. Adultery is not a crime in Washington nor is it recognized as relevant in divorce proceedings. i.e. Washington is a “no fault” state.

Caveat: I’m not a lawyer.

Hence, "in some states".

It's entirely possible some form of pre/post-nuptial agreement exists, as well.