A quick search shows that one of the meanings of the word "designer" is as an adjective or, more accurately, an attributive noun (see @Mary-Lou A's comment below, and this article) and is something like this:
To have been made by, or to have the expensive sophistication of, a famous and prestigious designer.
But there is a variation on that precise meaning which might apply in this context. It is something like this:
Appearing to have been made by, or to have the expensive sophistication of, a famous and prestigious fashion designer.
In other words, it might not be unreasonable, in this context, to interpret "designer" as a form of marketing BS; what the Wikipedia Manual of Style might call puffery or a "peacock term".
Note that I say only that it might be marketing BS. For all I know, the birdhouses in question were designed by Michael Kors, influenced by Frank Lloyd Wright, and have four floors, twelve bedrooms, pool, servants quarters, and grand entrance hall complete with gold-trimmed furniture and a swing.