The tester is one of those microcontroller based component testers. It identifies the component and shows the pinout.
So?
I guess you need a little more information. This tester is not the nearly worthless tester that you find on some multimeters. I've found that it does a good job of identifying components. It's intended to be used with simple two terminal parts like passive components or, three terminal devices like bipolar transistors triacs mosfets etc. When you plug in something unusual it commonly shows it as an unknown or failed part. Sometimes it will identify it as a combination part like a BJT with a flyback diode across the collector and emitter. It's a useful test instrument. Very useful to me since I have a load of old no number or house numbered parts. It's nice to be able to plug in a bunch of unknown transistors and quickly find their polarity and gain.
Also, there is a thread for this tester on the eevblog board.
https://www.eevblog.com/forum/testgear/$20-lcr-esr-transistor-checker-project/
The tester is current limited and shouldn't kill the lm34.
then nothing to worry about.
I've been working with electronics for over 30 years. I've learned when to be cautious. I'm very cautious when I have no spares for a needed component.
I'm currently finishing the code for a PIC 12F683 based temperature controller. I'm at the point where I'm tweaking up the thermal response of the system. Without a working LM34 my tweaking will be on hold. With my last LM34 on the breadboard, I'm being really careful with it. I debated even testing the LM34 with my multimeter diode check function. Since I know that the multimeter diode check is limited to 1mA current and 3.5V open circuit voltage I decided that this was fairly safe.
Plugging my last LM34 into my transistor tester is an unknown. I know that the tester has current limiting. No idea how much current or the open circuit voltage.
I should have some more LM34's in a day or so. I'll then plug one into the tester and post the results.
Finally, back to the main point of this thread. The fake LM34.
As others have said, I'm also surprised that the Chinese would bother to make fake parts that sell for about $1.
Maybe low price fake parts are just a way to sneak past the EBay checks. If you sell a few $500 fake parts every buyer will be complaining to EBay. With cheap parts normally bought by hobbyists, the buyer might just assume that they made a mistake and destroyed the part. The buyer also might not bother to get a refund or return for something that only cost a few dollars.
With a bunch of EBay sales like this you could make good money even on parts that only sell for a dollar!