
Later, I found out that it was a limited trial (48 our of approximately 310 TK Maxx and HomeSense stores are currently taking part, according to https://www.bigbrandtreasure.com), so when I tried to use it at HomeSense in York, I wasn't able to. However, purchases from non-participating can (apparently) be registered onto your account online.
The card registers you into monthly prize draws to win gift cards (you get entered once for doing nothing, and every purchase gains you an extra entry), qualifies you for cardholder-only events (after three purchases in three months), and enters you into competitions to win experiences (after six purchases in three months). There are also cardholder "benefits" every three months.
Overall, it does seem a bit of a naff scheme for casual shoppers, but I doubt they're the target audience.