fürstlich
lavish
(Figurative) Extremely generous, luxurious, or elaborate; in a way that recalls the wealth and splendor of a prince.
Bei der Hochzeit wurden die Gäste fürstlich bewirtet.
At the wedding, the guests were lavishly entertained.

princely / lavish
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lavish
(Figurative) Extremely generous, luxurious, or elaborate; in a way that recalls the wealth and splendor of a prince.
Bei der Hochzeit wurden die Gäste fürstlich bewirtet.
At the wedding, the guests were lavishly entertained.

The adjective 'fürstlich' (princely/lavish) is derived from the noun 'der Fürst' (prince). It means either literally 'belonging to a prince' or, in a figurative sense, 'luxurious and generous like at a prince's court.'
Der König empfing seine Gäste in den fürstlichen Gemächern.
The king received his guests in the princely chambers.
You can recognize the meaning from the context. If the word refers to historical people or buildings (e.g., castle, family), it is meant literally. If it describes an action or a thing (e.g., eating, salary), it is meant figuratively in the sense of 'luxurious.'
Sie erhielt ein fürstliches Gehalt für ihre Arbeit. (figurativ)
She received a lavish salary for her work. (figurative)
In the figurative meaning, 'fürstlich' (lavish) is often combined with verbs of enjoyment or reward: 'fürstlich speisen' (to dine lavishly), 'fürstlich leben' (to live lavishly), 'fürstlich entlohnt werden' (to be paid handsomely), or 'jemanden fürstlich bewirten' (to entertain someone lavishly).
Nach dem Lottogewinn konnte er endlich fürstlich leben.
After winning the lottery, he could finally live lavishly.
royal
'königlich' and 'fürstlich' are close in meaning because both bring to mind rulers, wealth, and splendor. Literally, 'königlich' refers to a king or queen, while 'fürstlich' refers to a prince or princess. Figuratively, both can mean luxurious or very generous. The difference is often in typical usage: 'königlich' works well for splendor, enjoyment, or a royal family. 'fürstlich' is often heard with very generous pay, a reward, or hospitality. They can be confused because both sound like high nobility and luxury.

Für die kurze Hilfe bekam Max eine fürstliche Belohnung.
For the brief bit of help, Max received a lavish reward.
Bei der Feier fühlten wir uns mit dem großen Menü fast königlich.
At the celebration, the large menu made us feel almost royal.