Must CA property managers be licensed real estate brokers?

Yes. One needs a real estate broker's license to manage another person's property. Anyone who seeks to sell, list, auction, or rent/lease real estate in California must be a licensed real estate broker or agent working under the supervision of a broker. However, if you are managing your own home or your own property, then neither a salesperson or broker license is required.
Specifically, California Business and Professions Code section 10131.01 provides:
"(a) Subdivision (b) of Section 10131 does not apply to
(1) the manager of a hotel, motel, auto and trailer park, to the
resident manager of an apartment building, apartment complex, or
court, or to the employees of that manager, or (2) any person or
entity, including a person employed by a real estate broker, who, on
behalf of another or others, solicits or arranges, or accepts
reservations or money, or both, for transient occupancies described
in paragraphs (1) and (2) of subdivision (b) of Section 1940 of the
Civil Code, in a dwelling unit in a common interest development, as
defined in Section 1351 of the Civil Code, in a dwelling unit in an
apartment building or complex, or in a single-family home, or (3) any
person other than the resident manager or employees of that manager,
performing the following functions who is the employee of the
property management firm retained to manage a residential apartment
building or complex or court and who is performing under the
supervision and control of a broker of record who is an employee of
that property management firm or a salesperson licensed to the broker
who meets certain minimum requirements as specified in a regulation
issued by the commissioner:
(A) Showing rental units and common areas to prospective tenants.
(B) Providing or accepting preprinted rental applications, or
responding to inquiries from a prospective tenant concerning the
completion of the application.
(C) Accepting deposits or fees for credit checks or administrative
costs and accepting security deposits and rents.
(D) Providing information about rental rates and other terms and
provisions of a lease or rental agreement, as set out in a schedule
provided by an employer.
(E) Accepting signed leases and rental agreements from prospective
tenants.
(b) A broker or salesperson shall exercise reasonable supervision
and control over the activities of nonlicensed persons acting under
paragraph (3) of subdivision (a).
(c) A broker employing nonlicensed persons to act under paragraph
(3) of subdivision (a) shall comply with Section 10163 for each
apartment building or complex or court where the nonlicensed persons
are employed."
If the property "manager" in question fails to fit into these exceptions, you may have grounds to take action to rescind your lease. However, I think it is a very risky position to take to withhold rent on this basis.

Business and Professions Code section 10130 declares it is unlawful for any person to engage in the business of a real estate broker without holding the appropriate license. Business and Professions Code section 10131 specifies that a real estate broker includes a person who "for compensation or in expectation of a compensation, regardless of the form of payment . . . (b) Leases or rents of offers to rent . . . . leases on real property . . . or collects rents from real property . . . ."
A prior answer has pointed out various exceptions to this law. However, the law itself contains an exception in that it does not prohibit all power-of-attorney or agency relationships that are not taken on for compensation. For instance, an owner might grant a power-of-attorney, for no compensation, for a relative to enter into a lease of a home.
From your description, it appears that you believe the manager is acting illegally and without any exception. This may give you various rights, including a right to seek rescission of the lease as previously mentioned.
But to withhold rent and rely upon this as a technical defense to an unlawful detainer is probably a dangerous strategy. An unlawful detainer might be brought by the owner, and unless you prove your habitability defense or another reason why you were not required to pay rent, the court may not be sympathetic to your non-payment of rent due to an unlawful management arrangement.
Your question seems to suggest that a tenant that leased through a manager without the required real estate license could remain in a rental unit indefinitely, without paying rent, regardless of whether there was a valid habitability defense or not. The court may not support such a position.
Sincerely,
Fred Hayes, Attorney
and Real Estate Broker