Old English had a letter called thorn that would eventually be replaced with the modern th or a y (ye olde is just said the old).

The y^e form was apparently used for the King James Bible in 1611 so it's plausible it might still have been a normal way of writing "the" as late as 1650, rather than as an affectation or abbreviation.

Edit: He also uses y^t for that which I hadn't noticed at first glance.

You could raise the same question about many historic ligatures, like abbreviating "et" to "&" in Latin?