Indeed they r related, but the relation can't be merely understood from a layman's perspective.
Let's take the example of Ràm Naam.
राशब्दोच्चारणादेव मुखन्नीर्यन्ति पातकाह्।
पुनः प्रवेशभीत्यात्र मकारस्तु कवाटवत् ।।
When u open ur mouth to say "Ra" the sins will go out & when u say "m"
and close ur mouth so they can't enter back.
Now there is a technique followed where u take ur sins to trgh the right nostrils up to navel point and breath (its contempleted). Then the sadhaka assumes that his inner fire has burnt all his sins. Now the ash that is left is thought to be removed from the left nostril. (The procedure was quite complex even though I don't remember. In case I find the complete procedure will update it here.)
So, u see what's striking is that in both ways the sin is thrown out. When u say Ràm the air goes out from the mouth, same as in that breathing technique, n in both ways the sins leave.
Now was that related to breathing?
Yes, it was but when u look into other paths, where this breathing technique is followed in Karma yog. And naam jaap is more concerned in layman's terms to Bhakti yog. So, tbh it isn't that sound to link two far-spread branches.
Apart from it, now, isn't it wonderful that, what u could have to do all by yourself, like imagining a fire, burning the sins, then thinking of ash n throwing it from nostrils. That's far more complicated, right?
Imagine,u got the same fruit by a mere jaap of the word Ràm.
So, have faith in the name despite to what Sampradaya one belongs. Be it Shakta, Shaiva, Vaishnava, or any. Because every deity got a Sahasranama, n thus the namjaap has a special place despite differences in sampradayas