Note that while he had limited support (mostly in the sense of "sure, we'll give you a place to stay and a small retainer as a bargaining chip against the English) from each, it wasn't simultaneous; he bounced around from one court to another depending on their relationship with the Tudor monarchy (since Henry VII obviously made "don't harbor my enemies" a key demand for any alliance). When England was aligned with the Habsburgs against the French, he was welcome in France; when England made peace with the French, he went to Burgundy (where Margaret of York was willing to support anyone who would bother the Tudors) and eventually (after a failed rising in Ireland) to Scotland (which also had a history of feuding with the English and wanted a bargaining vhip).
The linked blog massively overstates the case, by the way (for example, while it's true Henry VII used Warbeck as an excuse to get Parliament to approve taxes, he then kept those taxes and used them for domestic development when the threat passed, which was not the first time he pulled that trick, and one reason he was one of the few English monarchs to leave the government in sound fiscal shape when he died). There's no evidence anyone actually believed him to be Richard, Duke of York, rather than a convenient tool to threaten the English government; note that his various conspiracies got significantly less support than the blatantly fraudulent Lambert Simnel (admittedly, that's partially because the remaining discontent Yorkists had mostly shot their wad at Stoke, but it's worth noting that none of his foreign backers was willing to spring for a few thousand mercenaries for Warbeck).
More broadly, it doesn't matter who he is, it matters whether he can convince anyone (which mainly depends on whether he can win the throne, which seems increasingly unlikely by the time he shows up, as Henry VII has had enough time to solidify his rule).