Yes, they do. In many cases, hardware cards (especially sound cards) came with their own device driver files - .SYS (usually) files that were loaded and initialized at boot time by being included in CONFIG.SYS.
The hardware could use the upper 384k of address space (above 640k) to address its own built-in RAM or ROM, and could install memory-resident code that hooked into system calls, so that application programs could use the new hardware by making standard system calls.
This was not the same as a TSR.
TSRs were conventional applications (.EXE, or, more commonly, .COM) that could leave chunks of memory-resident code hooked into the system interrupt calls, much like a device driver, using the standard “Terminate and Stay Resident” system call, but had to use undocumented tricks to avoid re-entering non-reentrant system calls when triggered by interrupts.
(DOS had no native provision for multitasking, but included a multitasking print spooler that used those undocumented hacks to prevent problems. We all learned how to make TSRs by disassembling the print spooler and figuring out its tricks.)
There’s some overlap - some devices used TSRs as device drivers, and some device drivers were .EXE files, but device driver files for specific add-on hardware devices were quite common.
