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These are electric cars that is cars that run on electricity.
or
These are electric cars that are cars that run on electricity.
To clarify cars that run on electricity does not add any new information, it merely defines/clarifies what an electric car is.
Sorry if this is stupid I've actually never heard people use that are for definitions.
The phrase 'that is' does not change when introducing plural things. You could consider it to be short for 'that is to say'.
Carers, that is people who are providing unpaid care to a family member or friend
The construction of "that is" in this sentence is an ellipsis used in a clarification rather than a use of a relative pronoun.
There are electric cars, that is to say, cars that run on electricity.
It clarifies what is meant by the term "electric car" and may get shortened to
There are electric cars, that is, cars that run on electricity.
It is of course possible to use a relative clause to clarify meaning. So you could say
There are electric cars that are cars that run on electricity.
Notice, however, the parenthetical "that is" is set off by commas, which reflect a slight pause in spoken English.
However, as you point out, the whole sentence is quite clunky because it is redundant.
Electric cars [obviously] run on electricity.
EDIT: Because a sentence seems to be poor English, people frequently ask whether the sentence contains grammatical errors without considering whether the sentence may be clumsily constructed. This question is an example. You wondered about the grammar because your ear for English correctly suggested that it was a terrible sentence. But not all terrible sentences are ungrammatical. It is quite possible for bad English to be flawless grammatically.
