Different drivers/support for 10GbE SFP+ copper versus fiber?

Would they really have/need a different SFP+ driver if the connection is copper instead of fiber?

They might.

Direct-attach copper (DAC) connections use twinax with unchanged PCS encoding (64b/66b for 10GBASE-R), the same that's used for multi-mode (-SR) or single-mode fiber (-LR). Accordingly, the device could completely ignore the link details, but it might not.

SFP modules provide type and vendor identification, and possibly diagnostics via I²C. A vendor lock is quite common - a device accepts only "original" modules and DACs (or modules claiming such - there's a vast third-party, compatible market).

10GBASE-T is a different flavor as it's got it's own, very elaborate PCS code. Therefore, 10GBASE-T SFP+ modules either feature a transcoder chip for -R to -T (not accepted in all devices) or they are fed a proprietary pre-cooked PCS code from the device (device and transceiver types need to be matched specifically). The former requires considerable power, limiting the reach of the twisted-pair to usually 30 m. The latter is highly proprietary and mileage varies. 10GBASE-T is also increasingly picky on the cabling, so it might not be your best choice.

So, you should make sure whether your devices accept generic modules ("MSA standard") or a certain brand/type only. For twisted-pair aka RJ45, make sure the SFP+ port is compatible with 10GBASE-T transceivers. With DACs there's the added complication that they've got two ends which must be compatible with both devices simultaneously. There are custom-made "Frankenstein" DACs with different ends, but they're third-party only and can be cumbersome later on unless you can program them yourself.

For simplicity, I'd recommend using 10GBASE-SR transceivers with multi-mode fiber. They have decent reach, can be matched across vendors and are available for just about any device around. If you're going the 3rd-party way, you can just as well use single-mode and -LR transceivers - the price is about the same (beware of original parts markups though!), and the cabling is more future-proof.