Could a linear power supply make a difference on a high-quality raspberry-pi based streamer sending signal to a high-end DAC using an I2S connection?
Almost anything
could make a difference but
not because it's switching, linear, or a battery.
A power supply just has to put-out a DC voltage and it has to be capable of supplying the necessary current. In most cases we want a reliable-regulated voltage that doesn't have
noise* and that doesn't vary with the load or with normal variations in power line voltage.
Analog audio (and your ears) are very sensitive to noise.
But
digital electronics are highly immune to noise. Most computer power supplies are too noisy for analog audio but with billions of bytes being moved around and processed in the computer "random errors" almost never happen. If the data gets "damaged' your digital audio will be damaged but that almost never happens.
The most common issue on the digital side is if the audio gets interrupted and you get a glitch in the sound. That's most-often related to multitasking and unrelated to the power supply. (If you are not working with real-time audio or video an interruption or delay isn't a problem.)
* Electrical noise is high frequency variations (higher than DC which is zero Hz) and the noise rides on top of the DC. With a regular linear supply it's usually 100 or 120Hz (double the line frequency) and with switching supplies it's usually well-above the audio range. Being beyond the audio range can be an advantage in case any noise gets past the power supply's filters.
In computers all of the logic-data switching feeds noise back-into the power supply and you can get lots of noise in the audio range. That is sometimes a problem with UDB powered audio interfaces if the noise from the USB power gets into the analog circuitry.