Learn Spanish (Latin America): Observo el modo en que mezclas la ginebra con el licor. - I am observing the way you mix the gin with the liquor.

Because the action is a known reality happening right now. The speaker is actively watching the person mix the drinks. If they were talking about an unknown or hypothetical future action, like 'I will observe the way you mix it tomorrow', you would use the subjunctive mezcles.

Could I say el modo en el que instead of just el modo en que?

Yes, both are perfectly correct. With these relative clauses of manner, it is very common to drop the definite article right before que. Saying el modo en que is a bit more concise, but el modo en el que means the exact same thing.

Are la manera and la forma interchangeable with el modo in this sentence?

Absolutely. You could say la manera en que or la forma en que instead, and the meaning ('the way in which' or 'how') remains exactly the same. They are all great options to have in your vocabulary.

The English translation says 'I am observing', but the Spanish uses the simple present Observo. Could I say Estoy observando?

You definitely can use estoy observando to emphasize the ongoing nature of the action. However, Spanish frequently uses the simple present to describe things happening right now, whereas English relies heavily on the present continuous ('I am doing'). Both sound natural here.