No, it is not inherently rude to give another adult a birthday card with no money in it. Birthday cards are a thoughtful gesture and a way to show someone that you remembered their special day and took the time to express your wishes and sentiments.
The value of a birthday card comes from the thoughtful message you write inside and the effort you put into selecting or creating the card. It's the sentiment and well-wishes that count, not the monetary value.
Many people appreciate heartfelt and sincere birthday cards without expecting any money or gifts. However, cultural norms and individual preferences can vary, so it's essential to consider the person's personality and your relationship with them. Some people may prefer money or gifts as part of their birthday celebration, while others may place more value on the thought and message behind the card.
If you're unsure about what the person might prefer, you can always add a warm and personal message in the card, expressing your best wishes and appreciation for them. If you feel compelled to include something more than just the card, you could consider other thoughtful additions, such as a small gift, a gift card to their favorite store or restaurant, or an invitation to do something fun together to celebrate their birthday.
Ultimately, the key is to show genuine care and thoughtfulness in your gesture, regardless of whether you include money in the birthday card.