No, that isn't a Christmas cactus. A true Christmas cactus has scalloped leaves, and are hard to find. My nephew has my mother's, and it's probably 40+ years old. If you Google "Christmas cactus vs. Thanksgiving cactus" you can see the difference.

You are showing Thanksgiving cactus (Schlumbergera truncata). Thanksgiving cactus has pointy, claw-like leaf edges. Flowers point outward. Yellow anthers. Bloom time October to November.

The true Christmas cactus is Schlumbergera × buckleyi (hybrid of hybrid of S. russelliana x S. truncata). It has smooth, scalloped (rounded) leaf edges and pendulous flowers that tend to hang downward. Purple or brown anthers. Bloom time late Nov into January.

To further complicate things, Easter cactus also has smooth rounded leaf edges. Flowers are more star-shaped and point upwards. They belong to the genus Rhipsalidopsis. Bloom time late winter, early spring.

Most of the ones you see available at big box stores are the Thanksgiving ones, followed by Easter sometimes. Christmas are harder to find.

But the term 'Christmas Cactus' is often used informally by the novice to refer to any of these three different plant types.