I guess the easiest and fastest way would be to slap on tape.. but bestest is sewing. plus removal is still pretty easy with a seam ripper.. if you really need it off. Im thinking in quality and the fan base noticing details requires them to sew these on.
and my only objections to the double sided tape idea is that 1) it would not even stay as attached as long as an iron-on or melty glue..
because its fabric, that tape just comes off with minor abrasion or moisture or even in the context of spfx.. adding wind. and 2) the thickness again would still be about 1/16 or so and you would have to keep the tape the same shape as the patch for you not to see lump or ridge or edge, or have the edges seem to float and be caught by.. wind. Also tape has been used on harder surfaces and logos on uniforms in the past year or two in the Major Leagues and it has shown that over the course of a couple games, it has the tendency to twist or be torn off or rubbed off with a bit of use.
there is super thin double-sided fabric tape out there, but again that is heat pressure applied, and only meant to be temporary until you sew it as a permanent solution. It tends to release over a short time if you dont get enough heat or pressure on it. plus its not that great of a glue to start with.. its a lot like weak iron-on patch backing and heat application on leather is a no-no.
I sew a lot of stuff.. from sports jerseys to creating costume pieces to military jackets and bags and liners ... it just feels like the best way to attach these things is with a quick pass on a machine. if they use invisible thread on top and keep it next to another embroidery seam or detail it would not be seen. and if you keep the correct tension you wont notice any tell tale spotting or the pull of the thread thru the holes.
Another strike against the tape is the contours and folding of the jacket sleeves. wrapping around the arm and
constant movement and folding of the bends in the sleeves make for a sewing need in my opinion.
If "this is Sabine's jacket" or "Carson's jacket" then you really don't need to take off the patch for using the jacket for some other character, you just
get a different jacket or use it in the background and hope no one notices its actually Sabine's. I'm sure she keeps her own wardrobe
and they don't reuse it for others. Otherwise they have to keep adding and subtracting these patches for swap outs.
then you have people like us noticing those things and saying how dumb they are for doing that.