Are radio waves naturally polarized like light or is that a function of how they are produced?

Radio waves and light are both electromagnetic waves. The only difference between them is the wavelength.

The polarization is initially determined by the way they are produced

FM Broadcast radio is generally transmitted by circularly-polarized antennas in order to accommodate receivers in most any orientation.

AM broadcast towers are vertically-polarized.

The propagation tends to be by ground waves so retain their polarization for the most part. You may remember the old transistor radios used ferrite loop antennas that had a particular polarization, and you could move them around to find the best angle for reception at the best strength for the receiver's circuitry.

Radio waves emitted by an antenna have a specific polarization, and receiving antennas are generally sensitive only to a specific polarization. So in principle if the transmit antenna were strictly vertical and your receive antenna were strictly horizontal, you would receive nothing. But there are a couple of complexities:

• Partially-aligned linear radio antennas can receive each other with modest losses.
• A circularly polarized antenna can receive any linear polarization with modestly reduced efficiency, and vice versa.
• Short-wave signals are generally received after bouncing off the ionosphere, which randomizes the polarization. Similarly, Wi-Fi and other 2.4/5 GHz signals are often bounced off buildings or walls, which tends to randomize the polarization.
• Signals that are not narrow-band can have complex mixtures of polarizations, and polarization can change very rapidly with time.

The key difference between radio waves and visible light is that most of the radio signals we are familiar with are produced by coherent emission processes, which (usually) produce fully-polarized radio waves. More, almost all detectors of radio waves coherently detect just one polarization; radio astronomers usually use pairs of crossed dipoles so we can record both polarizations and reconstruct the input signal's polarization state.

Most of the visible light sources we deal with are incoherent and produce unpolarized light (an even mixture of polarizations) and our detectors mostly aren't sensitive to polarization anyway. Lasers are coherent and indeed are polarized, but unless the laser is designed to have a stable polarization, you tend to get random jumping around on very short time scales, averaging out to unpolarized. The human eye is in fact very slightly sensitive to polarization, though we don't usually pay attention, and there are processes - like reflection - that readily add polarization to light, hence the utility of polarized sunglasses (to preferentially block light reflected off horizontal surfaces).

The polarization of the emitted wave is a result of the antenna design.

Typically, antennas that primarily extent in vertical direction have vertical polarization. However, it's often hard to see the actual antenna inside the protective cases they come in, so, if you see an antenna thing that higher than wide, please don't assume vertical polarization. It might just be one case with many stacked smaller antennas inside, or a loop, or something else completely.

And yes, the polarization of the receiving antenna must match the polarization of the wave, or else you get worse reception.