In the IC processes I have worked with, the fuses which are used for OTP memory are made from polysilicon.
They're simply polysilicon resistors with a value of a few hundreds of ohms when intact. After such a fuse is blown the resistance increases by a factor 100 or so to at least 50 k ohms if I remember correctly.
It is mentioned in the comments that metal can be used for the fuses and that is true, see this article.
So the OTP fuses can be made from polysilicon or metal. Other materials might be possible as well.
Is there a difference between how they are implemented between low-power cheap microcontrollers (which might have a few OTPs for lock bits) and high-end x86 chipsets (with potentially thousands of OTPs to store manufacturing settings and even entire RSA keys)?
There probably is, the "fuse based" OTP memory is quite "expensive" as it needs a lot of silicon area per stored bit. On a microcontroller chip where there is already flash memory available it might be easier to implement the OTP lock bits using flash memory as well. The logic circuits then simply prevent further programming when a lock bit prevents it. That way no fuse is blown but you cannot program the chip anymore.
A similar scheme can be used for manufacturer programmed settings, once set the chip prevents the settings to be changed.