What is an electric toothbrush?
"An electric toothbrush is a toothbrush that's bristles run on electric power."
Is this ungrammatical English.
Is that sentence ungrammatical English?
An electric toothbrush is a toothbrush whose bristles run on electric power
'whose' is the possessive relative pronoun. Not that's, who's, or which's
<<An electric toothbrush is a toothbrush whose bristles run on electric power.>>
A toothbrush is not a person, hence we shouldn't use "whose" there unless for some reason you're personifying the toothbrush.
Using "whose" there actually sounds silly to me, as if the toothbrush is a person. "which's" actually makes more sense.
"An electric toothbrush is a toothbrush which's bristles run on electric power."
No, "whose" can be used for either people or things. "Who" can't, but "whose" can.
There is absolutely nothing wrong with using 'whose' for things
Longman Dictionary of Contemporary English
definition 2 Used to show the relation between a person or thing and something that belongs to that person or thing.
The examples it gives are
That's the man whose house has burned down.
A new laptop computer whose low cost will make it attractive to students.