The short answer to this is that, immediately after DNA replication (S phase), you are right: there are 46 chromosomes and 96 sister chromatids. Let's break this down further.
First let's consider mitosis.
- Before DNA replication, imagine that there are 46 chromosomes, 23 red (paternal) and 23 blue (maternal), each with just one sister chromatid
- After DNA replication, there are still 46 chromosomes, but now 92 sister chromatids -- these are the classic "X" structures of condensed chromosomes that we typically envision
- During mitosis, the two sister chromatids on each chromosome are split apart such that each daughter cell ends up with 46 chromosomes (each with just one sister chromatid)
Now let's consider meiosis (see image in reply).
- Before DNA replication, there are 46 chromosomes, 23 red (paternal) and 23 blue (maternal), each with just one sister chromatid
- After DNA replication, there are still 46 chromosomes, but now 92 sister chromatids
- During meiosis I, the homologous chromosomes are split apart so that each daughter cell ends up with 23 chromosomes, some red and some blue (each with TWO sister chromatids, so 46 total)
- During meiosis II (which is analogous to mitosis), the two sister chromatids on each chromosome are split apart such that each daughter cell still ends up with 23 chromosomes, but each with just one sister chromatid