The SPI statistic can be considered a standard normally distributed random variable $SPI \sim N(0,1)$. Thus the expected value, or mean, of the SPI statistic is $0$, and its standard deviation and variance are $1$.
If the mean of the distribution is zero, then by the central limit theorem, you would expect that the limit of your sample mean (the mean of ~40 years of data) would approach zero as your sample got larger. So yes, it is both possible and expected.
What may be causing your confusion is that SPI can be calculated for different time sets. If you are analyzing month by month, then I would assume that your dataset is divided by month. In that case, the SPI is calculated individually for each month, so that the mean of any month or set of months over many years would tend to zero, as per the last paragraph.
It would not make sense to calculate a monthly SPI from a 12-month mean divided by 12 (to give you a mean per month on an annual basis) In seasonal wet-dry climates you would have massively negative SPI value in the dry season, and highly positive numbers in the wet season. The monthly means are individually given an SPI statistic to allow apples-to-apples comparisons with the same month in other years.