Is my hen and chick flowering? I see the stems are purple.

Nope... not flowering.
They can send up a stalk when producing flowers.

When I searched the pic with google lens, a number of cultivars came up with similar colouration.

Maybe compare to SEMPERVIVUM 'Ruby Heart'?

Those are the florets, the plants themselves, it doesn't show any stems.
Here is our generic sempervivum entry. Scroll down to see all the member submitted photos with various colors of plants. Several show the flowering stems that shoot up from the centers. Also, there are a large variety that change colors in the season or with temps, and some varieties come in these shades.
Sempervivum

@Zplant
Welcome to NGA!

That's a beautiful well-grown Sempervivum! The rosettes you are seeing are actually individual leaves clustered around the plant's center. These rosettes are the "hens". They might produce smaller rosettes on stems that grow out from between the individual leaves. These would be "chicks".

The coloring of the leaves can change, especially between the seasons, as mentioned above by Sue.

Pat