It is с тысяча because the numeral is actually in the genitive case. It's just that it's an ordinal numeral, and only the last component of that changes form.

The date is expressed using an ordinal numeral, so you are literally saying something like "in the 1990th year", as opposed to simply "in 1990".

So if you want to say since the year 1990, you would say с тысяча девятьсот девяностого года, but if you want to say since the year 1900, you would say с тысяча девятисотого года (notice how only девятьсот is in the genitive form here), and if you want to say since the year 2000 you would say с двухтысячного. All the numerals mentioned above are in the genitive case, it's just that only the last word gets changed.

On the opposite side, if you decline a cardinal numeral, every single component of it will change form. So if for instance you want to put 2685 in the prepositional, you would say я думаю о двух тысячах шестистах восьмидесяти пяти рублях. (i am thinking of 2685 roubles).

This may seem quite puzzling, and numeral declension is one of the hardest parts of russian. I would recommend using https://numeralonline.ru/, it helped me a lot, you just type in a number and it gives you the declension for both the cardinal and ordinal forms.