@expat48 - GREAT QUESTION!!! Everything affects the tone A LITTLE BIT. The shape of the cover plates, the material of the comb.
Now, Richard Sleigh claims that he can hear the difference between a single lacquered pear wood comb in the Marine Band 1896 and the triple-lacquered pear wood comb in the Marine Band Deluxe. Like @Slim, I’d have to see him do a double blind test to believe that!
I do think that comb has a SLIGHT effect on tone because when the reeds vibrate they will bounce a bit off of the comb materials in the chamber before traveling out to listeners ears. As to whether this change in tone is indeed perceptible…it would be quite a bit of work to create satisfactory scientific experiments on this subject so in the meantime we’ll all have to indulge ourselves in our opinions.
But, undoubtedly, the MAIN THING that effects the tone of a harmonica is the Material, Shape, and Position of the Reeds - the objects that are generating the tone with our breath.
My Joe Spiers harps sound different than stock Hohner harps because of the way he works the reeds. It’s rather astounding.
Now that we’ve belabored shop talk about gear, LET’S GET TO THE HEART OF THE MATTER.
ALL HARMONICAS sound thin if you put them in front of a fan.
What makes any harmonica sound HUGE and WARM and BRIGHT in all of the BEST ways, is the way it resonates in our mouths and throats.
Note that Brendan Power makes Konsheng harmonicas sound AMAZING. (They don’t sound like that when I play them! 
)
So while we’re on this topic - Vowel shapes, throat position, and hands have even bigger impact on tone than reed materials, shapes and positions.
“Uh” or “Oo” like Book are the WARMEST sounds. Also, opening the back of the throat and lifting the soft palette are worth exploring in this regard!
“Ee” is the widest BRIGHTEST sound on the harmonica. Sonny Terry was a master of using this ultra-bright tone especially on the -4, without having to play super-loud, which Joe Filisko refers to as “Icepick Tone.”
And of course, closing the hands around the harp (or playing into a coffee mug, or the harp wah with hand covered) produces the warmest tone, and playing open produces the brightest tone.
And then of course we could go down the rabbit hole of mics, pedals, and amps.
@blues_harp_cat welcome to the forum and great catch! Like @Slim said, brighter tones are more harmonically complex waves with more overtones, and warmer sounds are simpler more fundamental waves, which is why if you listen to a sine wave it has a very mellow, warm tone.