Is making handmade holiday cards as a hobby and source of income considered an innovation in religion (bid'ah)?

The fundamental principle in dealings is permissibility and lawfulness, unless there is evidence to the contrary. The ruling on cards made for holidays varies depending on those holidays:

If the card is to congratulate on a disbelievers' holiday (such as Christmas), or an innovated holiday (such as the Prophet's Birthday), or holidays that involve imitation of disbelievers (such as national holidays and birthdays): then it is not permissible to make, sell, or buy them, because it involves an affirmation of those holidays and cooperation in sin and transgression.

If the holidays are legitimate (only Eid al-Fitr and Eid al-Adha): then it is permissible to make and sell greeting cards for them, provided that lawful or permissible phrases are chosen, such as "May Allah accept from us and from you."