Is an ampersand formal?

The ampersand should not be used in formal writing, except in special cases like names of certain companies, in which case it should be used if it is part of the name.

I think it's not so much a case of formality, but rather one of fashion, iconicity and stylistic preference.

Writing sequences such as "& Sons", or using the "&" between two surnames forming part of a company, has in some senses become an "iconic" use of the "&" that has tended to stick. It's also used in other cases that help an abbreivation "stand out", e.g. "M&S" ("Marks and Spencer"), "D&D" ("Dungeons and Dragons"). Another argument for the "&" is that it represents "and" in contexts where it is 'obvious' that it would be present but one doesn't want to detract from the words either side, e.g. in writing "Mr & Mrs Daniels" on an envelope, or in references of the type "(James & Smith, 1988)".

But beyond that, the ampersand isn't actually used that much these days, in either formal or informal contexts. It's rare in modern media to need to save the space or typing time of a couple of measly letters, and an abbreviation such as "&c." for "etc" appears to have gone out of fashion, possibly alongside the other fashion of reducing the amount of "punctuation clutter" that people use in writing generally.

Strictly speaking, the ampersand is acting as part of the company name, so it is not really being used to replace "and". So this is not a "back door" to introduce ampersands to formal writing.