Learn Spanish (Latin America): Tú agitas que agitas la fórmula.

This is a special colloquial structure in Spanish used to emphasize continuous, persistent, or even slightly annoying repetition. While agitas y agitas simply means you shake and shake, using agitas que agitas adds a stronger, more dramatic sense of you just keeping on shaking it without stopping.

Do both verbs in this structure always have to match exactly?

Yes. In this persistent repetition pattern, the verb before and after que must be identical, matching in both tense and person. Because the subject here is tú, both verbs must be conjugated as agitas.

Could I use this exact structure in the past tense if I wanted to say you were shaking it persistently?

Absolutely. The structure works across tenses as long as both verbs match. You could say Tú agitabas que agitabas la fórmula to mean you were shaking and shaking the formula in the past.

If I wanted to replace la fórmula with the pronoun la to say 'You shake and shake it', where would the pronoun go?

You would place the direct object pronoun before the very first verb, and you generally do not need to repeat it. The sentence would become Tú la agitas que agitas.