Answer:
90 between 1000 and 9999
Step-by-step explanation:
The first digit cannot be 0. It can be nine other digits.
The second digit can be anything between 0 and 9.
To be palindromic the 2nd and 3rd digits must be the same.
And in addition, the 1st and 4 numbers must be the same.
So the answer is 9 * 10 = 90. That does seem too small, but try making a palindrome with the restrictions I have outlined. None of the examples you have given beginning with 8 are palindromic.
8118 is palindromic
8998 is also palindromic
9009 is palindromic
2002 is palindromic
There are only 90 of them.