My guess is that it may be protected by a patent. If you google, you see it is not widely available. So you can't propagate it to sell, but you can propagate for your own enjoyment.

There are many species of lavender - and some common names overlap, just to make life interesting (and confusing). Lavandula dentata is the French Lavender, and Lavandula stoechas is the Italian (or Spanish) lavender.

You'd have to be a keen botanist to be able to distinguish some of the cultivars that have been derived from any of the 'true' lavenders. I'm quite happy to know that the English Lavender (Lavandula angustifolia) has straight-edged leaves; that the French Lavender has dentated leaf-edges; that the Italian or Spanish or Greek Lavender has white flowers!

Names like 'Anouk', 'Grosso', 'Munstead' etc aren't really botanical names at all. They are registered names (possibly patented) of cultivars (cultivated varieties). I'm never interested in those names - I'm more interested in the botanical species. It would be different if I was a collector of lavenders and wanted as many different kinds as I could find.

If you want to propagate from your lavender (or any lavender) so that the offspring are exactly the same as the parent plant (cloning), it's very easy. Just take cuttings. For woody plants like lavender I prefer to take a heel cutting. From a long sprig like the one in your photo, you could probably take one heel cutting from the main stem, then cut off the top to make 2 or 3 stem cuttings.

As far as I know, it is not illegal to propagate from plants (unless the original plant is a prohibited one, of course). You can take as many cuttings as you like and fill your acres with the babies if you want. Also AFAIK, it's only the name of the cultivar that's patented - not the plant itself.

A patented plant is protected and cannot be legally propagated (by seed or cutting) for sale without having paid the licensing fee. Commercial growers have to pay a royalty for each plant in order to grow and sell them. Incidentally, not only is the plant itself protected by the patent but so is any sport or seedling that may be different from the patented plant. Technically, the genetics of the plant and any mutation is protected.

With a trademarked name only the name is protected and not the plant. There are no restrictions on propagating just using that name.

Sometimes a plant can be both patented and have a trademarked name.

Breeders do go after commercial growers who have illegally propagated their plants. They actually send out people to check growers and nurseries. Mine was inspected this year. And there are always stories in the trades about some grower getting caught.

Home growers are safe as long as they propagate for their own use and don't sell the plants.

Patented plants will often have p.p.a.f or p.p. somewhere on the label to indicate the restriction.(Plant Patent Applied For; Plant Patent)

Daisy is essentially correct about the common us of the names French, Spanish,etc.
However, these names have no botanical standing. L. stoechas is sometimes referred to as either Spanish or French. L. x intermedia (lavandin for short and including cultivars such as 'Grosso','Dutch','Provence') are also sometimes referred to as French lavender.

To further confuse the issue, "French" Lavender is grown in England and "English" lavender is grown in France.

Botanical names (including cultivar name) are the most accurate way to differentiate between plants. Especially when communicating with gardeners who may use different common names where they live.

I have found rosemary labelled as lavender at my local HD. So you aren't even guaranteed that you have what you think you have when you buy from them.