Normally you can't get achievements on minecraft demo but if you join someones survival world/realm (which you somehow can on demo), you can earn achievements.(My normal platform playing this version of minecraft is on Xbox)
MCPE-51773
Earning Achievements on Minecraft Demo
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Are you using the same gamertag to sign in to Xbox Live on the PC as you use when playing on Xbox? If so, this is working as intended. Achievements are linked to your Xbox gamertag; it doesn't matter what version or device you're using.
It does seem odd, though, that you got the demo version; if you own the game on Xbox, you should also own it on PC. Is this PC using a different Microsoft Account to sign in to the Store than the one your Xbox uses? If so, that would explain it.