Use of that

Both are grammatically immaculate.

Isn't the second "that" a relative pronoun referring to the first "that" which is a demonstrative pronoun ?

No, it's referring to 'bike', but so what if it did? It happens often: It's not that that I meant, it's this.

Is it not possible to write an emphatic cleft sentence with "that" as the subject and the relative pronoun coming later ?

What is your overall question here on this thread, DD-- or are you just rambling? Make your emphatic cleft sentence and I will check it.

What I wanted to know was whether we can use "that" as a provisional subject just as we use "it" in cleft sentence.
For example,
"It is a silly fish that is caught twice."
Can this be written as "That is a silly fish that is caught twice" ?

It is grammatically possible but practically non-native.

Thanks for your help Mister Micawber.
I have another question though regarding the cleft sentences.
"It's a fact that the Earth moves round the Sun."
Here "it" represents the nominal "that the Earth moves round the Sun"
Similarly "to+infinitive clauses" can also be represented by "it" as in "It's good to be home".
But, can we use "it" as the provisional subject for a single no

You will see it in informal English, with a comma:

It's a good movie, Hidalgo.
It's a great place, New York.