Confusion with applying incremental bandwidth to measure noise power spectral density
The sentence you're referring to describes a mathematical model; so, no, there's no bandpass filter for each \$\Delta f\$; that would be impossible, since \$\Delta f \to 0\$.

Describing it like this is necessary to mathematically capture the idea that if you measure something in arbitrarily small  regions, the thing you observe converges to a density; in your case, for any continuous spectrum, the idea is that your "power in measurable (hence, non-infinitesimally-narrow) bandwidth" converges to "power spectral density".