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"A tin of biscuits" vs "A biscuit tin" Is there any slight difference in meaning between that two phrases?

I'm not sure but I think there's a tiny difference in their meaning. Like, a tin of biscuits is about to emphasize that it's a tin with biscuits inside, while a biscuit tin emphasizes it's the bin, not the package.

Do you think it's so, or I'm just overthinking and the two phrases are the same, there's no, even a teeny-weeny difference in meaning?

No, you are not overthinking. There is definitely a difference, and the difference is the one you have described. A "tin of biscuits" refers to the biscuits within, whereas a "biscuit tin" generally refers to the tin itself.

Of course, if the tin is full of biscuits, then the two terms converge in practice. For instance, the two phrases:

"Please fetch me the tin of biscuits from the pantry"

and

"Please fetch me the biscuit tin from the pantry"

will mean exactly the same if the person requesting the tin is hungry and knows there are still biscuits inside! :)

As an aside, back in earlier decades (e.g. in the 1960s), people would collect the tins themselves (especially if they were very pretty) and re-use them to store other items.

I still have several biscuit tins from family members who have passed away (e.g. a biscuit tin full of assorted nails and screws from my grandfather and another full of interesting buttons from my grandmother). Even if I don't need the items within, I keep them for nostalgic reasons.

My mother in law has a biscuit barrel, these are still common

Hers is like this:

(they don't all look like little kegs):

Google Images (biscuit barrel)

I agree generally with the previous responses, but have one minor detail to add.

Semantically, the difference between "a tin of biscuits" and a "biscuit tin" can be significant. The first refers to an object, a tin, which contains biscuits, whether or not it was originally meant for that purpose. The second refers to a tin which was made to contain biscuits. Thus, a biscuit tin is still a biscuit tin, even after you have eaten all the biscuits!

Example 1: Look, Mom: Grandma sent me a tin of biscuits for Christmas, and they're my favorite kind. I do wish she hadn't sent them in this old tobacco tin -- yucky!

Example 2: Grandma, are you going to use that empty biscuit tin? I'd like to have it to store part of my bottle cap collection.

Compare it to A pack of crisps and a crisp packet. A box of shoes and a shoe box. a box of cigars and a cigar box. A barrel of oil and an oil barrel. A chest of treasure and a treasure chest, a barrel of wine and a wine barrel.

The first form states a substance, some items, associated with the container. The second form emphasizes the type of container, with or without it's contents.