We learned in Thor: Ragnarok that Mjolnir is only a focus for Thor's power, not the source. If so, when a worthy being lifts Mjolnir and gains "the power of Thor", where are they getting the power from?
In Thor (2011) there was the implication that Thor's powers came from an external source via Asgardian technology.
ODIN (CONT'D) You are unworthy of this Realm... Odin RIPS a disc off Thor's chest. ODIN (CONT'D) ...unworthy of your title... He rips away Thor's cloak. ODIN (CONT'D) ...unworthy of the loved ones you've betrayed. I hereby take from you your powers. Odin extends his hand towards his son. Mjolnir goes flying from Thor's grasp into Odin's hand. ODIN (CONT'D) In the name of my father... A FINGER OF LIGHTNING comes off the hammer and hits Thor, disintegrating the right arm of his armor and part of the chest piece. ODIN (CONT'D) ...and of his father before... Another strike disintegrates the remainder of Thor's armor, including the cape and torn-away disc on the floor.
But by Thor: Ragnarok (2017) it's clear Thor's abilities are not connected to any immediate source that could be taken from him (he's capable of lightning feats with no hammer).
My guess for the inconsistency would be that the hammer can contain or emulate the powers of Thor. So a person using Mjolnir would get the power of Thor from Mjolnir while Thor innately has the power of Thor. This might be a side effect of Mjolnir making his abilities easier to control.
ODIN (CONT'D) That hammer helped you control your power, focus it. But it was never the source of your strength.
Mjolnir itself is "magical" and was enchanted by Etri when it was created. One of these enchantments is the ability to summon and/or control lightning, thunder, and weather in general up to a global scale. "The Power of Thor" in this case simply means that whoever's worthy enough to wield Mjolnir will be capable of using the same enchantments as Thor would be. That's one of the explanations for the comic version during the original run of Thor, back when it was more fantasy-rooted than the more modern versions of the comic/character's continuity, the MCU included. I'm not sure when (if ever) it was established that Thor's power was simply focused through Mjolnir rather than part of Mjolnir itself outside of the MCU and they never quite explain whether or not Mjolnir is enchanted outside of always returning to Thor's fist. (I found this info on Wikipedia; I'm not a Thor expert by any means but I figured it might help. Cheers.)