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Note the term used sahasram or sahasra = 1,000 . It also means ' a very large number'
The term you offered is
सहस्राणि (sahasrAni).
Note the devanāgarī written form सहस्राणि is really sahasrāṇi. This informs us sahasra has been put in the plural form sahasrāṇi.

In the neuter gender, nominal case , 1st person, we have the following forms in 3 voices (vacana ) :
​singular: sahasram
dual : sahasre
plural : sahasrāṇi

Note I said in the nominal case, as the 'form' or declination also appears as sahasrāṇi in the vocative and accusative cases.

@sv. According to spoken sanskrit (http://spokensanskrit.de/index.php?beginning=0+&tinput=sahasra+&trans=Translate_), the meaning of "sahasra" (or "sahastra") is also 1000, large number, infinite, too many.
That means, sahasra can be less than 1000 as well, because for some people too many can be many decades or hundreds.

But this may not conclude anything. Since 1000 is the most objective among all the other meanings, people often choose it. But I do understand that, we cannot take 1000 as literal.