Can you use these interchangeably, for example:

ผมชอบอาหารแบบนี้

ผมชอบอาหารอย่างนี้

Pretty sure they both work, but is there a subtle difference, should one be used over the other depending on the context, subject, situation etc?

From Thai Reference Grammar

General Terms - อย่าง is for smaller units of things, like kinds of food, while แบบ is for larger units, or types.

เค้าสั่งอาหารสองอย่าง

เป็นวัฒนธรรมแบบไทย

This kind/that kind - These also mean 'like this/like that' and 'this/that style'. They can be put after nouns or used as the subjects of sentences. อย่าง can be used in two ways (also ยัง), the former is more formal.

wife says same