Verdict: False for voter-nominated offices
California's top-two system allows voters to choose any candidate for voter-nominated state and federal offices, regardless of the voter's party registration. In the general election, the two candidates who advanced from the primary appear as the choices, and a voter may select either one.
That remains true when both finalists prefer the same political party. The preference printed beside a name is information about the candidate, not a restriction on the voter.
The rule has boundaries
The top-two rule does not apply to the presidential contest, county central committees or nonpartisan local offices. Presidential-primary participation can depend on party rules, while nonpartisan contests do not nominate candidates by party.
A claim that California voters may always vote in every party contest at every stage is therefore too broad, even though the November claim is false for voter-nominated offices.
The quickest ballot check
Open the official sample ballot for the voter's address and identify whether the office is labeled voter-nominated, partisan or nonpartisan. That classification answers the rule more reliably than a campaign mailer.
Registration still matters for some primary procedures; it does not bind a voter to one party's finalist in a top-two general-election race.
Sources and further reading
California voter guide: Elections in California ↗