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For example, at the website of the Premier League, the zigzags of the lion's mane are used as backgrounds of the header and the footer; similar shapes are used as overlay masks on some photos.
More precisely, I refer not to allusions or resemblance of some elements, but their scaled reproduction that looks sike outlines, shadows, etc.
How is this re-use and repetition of certain elements or silhouettes called? What are other famous examples?
The re-use of similar shapes, patterns, spacing, type, colour and other elements throughout a design can simply be referred to as Repetition.
I would say that the Premier League example you provided demonstrates simple repetition. However there are terms for repetition at different scales which you might find interesting:
Self-Similarity refers to "a property in which a form is made up of parts similar to the whole or to one another" (Lidwell, Holden and Butler, 2010). Repeating design elements in this way, suggests an underlying order in a composition.
Self-similarity in natural forms - often at different scales - is sometimes referred to as Recursion. In the case of Recursion, an input is received, modified slightly and output as a similar form to the original. Trees, ferns, coastlines, and shells offer some examples of this. M.C. Escher demonstrated recursion and self-similarity in much of his work. Fractal art also demonstrates recursion.
M.C. Escher, Smaller And Smaller, 1956
Found example of fractal art (Possible source: matthewjamestaylor.com)
References:
In branding, it used to be common for designers to define a 'visual language' based on shapes derived or cropped from the actual logo. Arguably this is not so much 'in fashion' anymore, since large brands started minimizing their guidelines and most startups in recent years just went with typo-looking logos.
Vodafone are using that bubble in their logo as a placeholder for large headlines.
"Re-use and repetition of certain elements" is practically a dictionary definition of motif.
A repeated visual motif is used in many forms of design and art, and is often integral to a brand identity. Motif need not involve scale contrast or silhouettes though, nor does it have to be derived from the logo (although very common).
The most famous example I can think of right now is a pretty simple one: "Häagen Dazs" advertisements often use a malapropos umlaut in the tagline.
As soon as you start looking, you'll find motifs everywhere though. A couple other examples I found after just a minute of searches.
