Short answer: no. Buzzbassador can only track one referral code and one affiliate link per member, and a member can only belong to one program at a time. This comes up often enough that it's worth understanding exactly why, and what your two real workarounds are if you need something different.
One member, one program, one code/link
Every member account is tied to exactly one program. That program is what determines their commission rate, discount rate, and which referral code and/or affiliate link they have. Because a member can't belong to more than one program at the same time, they also can't have more than one active referral code, or more than one affiliate link, at the same time. One member always equals one set of rules, one code, one link.
This isn't a setting you can toggle on, it's how member accounts and programs are structured. There's no way to configure a single member account to carry two simultaneously-active codes for tracking purposes.
Why this comes up
The two most common reasons merchants ask about this:
Wanting to give one member a separate code for a specific campaign or promotion, distinct from their everyday code.
A member who feels like they're doing double duty for the brand (for example, promoting two different product lines) and wanting to track each separately under one account.
Your two workarounds
If you need to track more than one code for the same real person, you have two options:
Create a second, separate member account for that person. Since Buzzbassador tracks by account, not by legal identity, the same person can have two member accounts (typically with two different emails, or by using program routing to place them differently), each with its own code, its own program, and its own rewards tracked independently. Keep in mind this means two separate logins/dashboards for that person, and their rewards won't combine into one total automatically.
Create a second discount code manually in Shopify, outside of Buzzbassador. You can set up an additional Shopify discount code yourself and hand it to the member to use for whatever second purpose you need. Buzzbassador won't see, track, or attribute any sales made with that code, since it's entirely outside the platform, so you'd need to track usage and pay out any related reward manually.
Note: Moving a member between programs (for example, from one tier to another) is different from having two codes at once — that's a one-at-a-time switch, not simultaneous tracking. See How To Move Members to a Different Program Tier.