Your meter's "ºC/ºF" setting requires a K-type thermocouple which is sold as an accessory. Connecting your meter's "voltage" probes through alligator clips to the temperature probe likely produced one or more thermocouple junctions (Seebeck effect), and it could very well be that you "got lucky" in that your meter is responding to these junctions and not to the temperature probe itself. You might be able to verify this by removing the temperature probe, connecting the two alligator clips together, wrapping your hand (a heat source) around ONE of (not both of) the 'X' junctions shown in Fig. 1 below, and observing whether the meter still indicates a temperature reading (anything at all) when the "ºC/ºF" mode is selected:
DMM+ ---probe-->X<--alligator--\
#
DMM- ---probe-->X<--alligator--/
Figure 1
As @Whit3rd mentions, there are many types of devices that can measure temperature. Resistance Temperature Detectors (RTD) are also commonly used as oven/grill temperature sensing elements. When measured with an ohmmeter, a two-terminal RTD usually measures somewhere from tens of ohms to tens-of-thousands of ohms, depending on the probe's construction. For a probe the size of the one in picture you provided, the wire(s) that comprise the RTD's temperature sensing element could be very thin.
:: CAUTION ::
Some ohmmeters, when set to their low-ohms measurement range(s), can output substantial current. If an RTD is constructed from a very thin wire, that wire could potentially fail open circuit (like a fuse link that heats up and eventually melts or "blows") if too much current passes through the wire. If the wire fails open, that would explain the "O.L" overload indication you're seeing when measuring the probe's resistance.
And as others have already said, a thermocouple is basically two dissimilar metals that are bonded together at one end, with a temperature gradient across them. A two-wire (low precision) resistance measurement made with a handheld DMM across a thermocouple junction should indicate a "dead short" (~0 ohms). You're not measuring a dead short you're measuring open circuit ("O.L" overload): so either a) your temperature probe is not a thermocouple (it's an RTD, or a diode type, or whatever), or b) it is a thermocouple but it is damaged (failed open circuit). FWIW, thermocouples are typically very robust. One typically wouldn't damage a thermocouple by measuring its resistance with an ohmmeter--even one that outputs relatively high current on the low-ohms ranges; so I'm betting against option b) here.
