If I change my page theme, does the loyalty card on customers' phones change too?

Only if the card is set to follow it. On the Card design tab of the loyalty card screen under Marketing, the preset called Your theme colour re-colours the card whenever your storefront theme changes. Pick any other preset or your own colours and the card keeps them whatever the theme does. Saving a design tells you how many installed cards will update, and they refresh themselves within minutes. Any pair of colours under the readability threshold is refused, because a card nobody can read is worse than none. The Design & appearance tab in the docs covers themes.