The website http://www.grammarphobia.com/blog/2014/11/aunt-uncle.html says that the distinction is a matter of style, not grammar.
In the article, it quotes the Chicago Manual of Style, which seems to support the notion that in the phrase "your uncle Jack," uncle is not capitalized. But many people find fault with the CMOS, so you are free not to heed its dicta.
The site http://grammartips.homestead.com/betty.html makes an interesting distinction:
What about "her aunt Betty"? Well, that depends--if you mean "Aunt" as
part of Betty's "name," then write "her Aunt Betty"; but if you mean
it the same way you'd say "her sister Betty," then don't capitalize
But a Google Ngram of my aunt Betty vs. my Aunt Betty shows the latter more popular by a margin of about 2.5 to 1.
I personally would capitalize. To me, because most people call their aunts named Betty "Aunt Betty," aunt should be considered part of the name. Not so necessarily with other kinship words like sister or cousin, as mentioned above.
But ultimately, it's a matter of style. So I doubt you can reasonably say the e-card is incorrect; it would make just as much sense, though, if it said uncle Jack.