Yes, they are specific.
Every motherboard model can have different components on it, needing different drivers - the LAN port, the audio, video-capabilities, USB chipsets, memory controllers - all of these can and usually are different, model number to model number. So an HP Recovery Disk for one HP model probably won’t work for another HP model.
This is true for separate motherboards, too. Asus, MSI, ASRock, and every other assembler/mfr will provide a Drivers Disk for that one model.
Often, any attempt to cross-install results in a quick refusal (failure) and all the Name Brand folks have to do is insert a “works on X models” (listing any additional model numbers) and the Recovery Process shuts down.
Sometimes, I’ve been able to trick those IF I know what every motherboard-components’ version is. I mean, Dell, HP, Sony - they all buy from a finite set of suppliers (who seem to offer an infinite subset of products though!) - and often those component suppliers have their own drivers that can be installed separately. But it’s a chore and, on notebooks, it takes such a de-assembly process first, then re-assembly and testing to see what works and doesn’t.