TL;DR: I would analyze it as an adjective, but it's not a clear-cut case, and I think you can make a decent argument for analyzing it as an adverb.
The parts of speech are somewhat artificial constructs; they are an incredibly useful way to briefly describe the syntactic distributions of words, support the analysis of sentences and of words that they collocate with, and so on, but there are many edge-cases where a word has a distinctive distribution that does not fully conform to the usual patterns. When this happens, we want to find the best analysis — the analysis that accounts for the most observed evidence with the fewest exceptions, complications, and special pleading — but it's not realistic to expect that we'll always arrive at universal agreement about what part of speech a given word "is".
For example, in shone bright, some sources analyze bright as a predicative adjective (see e.g. the Cambridge Grammar of the English Language, § 5.2 (c), page 567), whereas others analyze it as a flat adverb (see e.g. https://www.visualthesaurus.com/cm/wc/when-adverbs-fall-flat/).
For bright red, similarly, both analyses exist; CGEL doesn't seem to explicitly address it, but it's easy to find other sources that give this either as an example of the adjective bright (e.g. http://www.dictionary.com/browse/bright) or as an example of a putative flat adverb bright (e.g. http://blog.writeathome.com/index.php/2014/01/the-parts-of-speech-102/).
We can sometimes tell whether something is an adjective or a flat adverb by comparing it with a more typical adjective with a derived -ly adverb, such as beautiful. So, let's try that.
We can easily say that a car has a beautiful red color, by using beautiful with an unambiguous noun:
The car is a beautiful red color.
The car is a beautiful shade of red.
The car is a beautiful red.
but if we try to say the same thing by modifying the adjective red, it doesn't work:
*The car is beautiful red.
[*]The car is beautifully red.
I think we can all agree that the first of these is clearly ungrammatical. The second sentence could be grammatical, but not, I think, with the intended interpretation: it doesn't mean that car's red color is beautiful, but that the car's being red is beautiful, or that the car is red in a beautiful way. (For example, if a car and a person's face are the same shade of red, then either both are a beautiful shade of red, or both are not (though in one case the color's beauty is less likely to be noticed); but it's conceivable that exactly one of them is "beautifully red", for example if we've got some schadenfreude about the person's emotional reaction.)
So, this approach doesn't seem to have worked: bright is neither like beautiful nor like beautifully.
So this may not be a straightforward case where we can point to one analysis as obviously correct. But, we can still try to come up with a good analysis.
Whatever analysis we come up with, we'd want it to be one that works for the many other similar words that are normally adjectives but that can modify color adjectives: light, dark, pale, deep, royal (blue), burnt (orange), dirty (blond) (hair), hot (pink), rusty (orange), yellowish (green), reddish (orange), and so on and so forth. Furthermore, we'd also ideally want it to account for why we don't seem to see any obvious adverbs used this way, and also for why don't say *"the car is beautiful red".
All this being considered, I'd propose something like this:
Color names can be used either as nouns or as adjectives. This applies even to such multi-word color names as bright red and navy blue, which have the internal structures Adj-N and N-N (respectively), but which can nonetheless be used as adjectives, as in "a bright red car" or "a navy blue dress".
where beautiful red doesn't count as a "color name" because the beautiful doesn't serve to identify the shade of red, merely to evaluate it. (Hopefully that's not too much special pleading.) Under this analysis bright is indeed an adjective, because it's modifying the noun red, even when bright red is an adjective rather than a noun.
But you may be able to come up with a better analysis, and perhaps that better analysis will handle it as an adverb.