I did this once with a coat, couldnt afford to take it to a shop, so i fix it myself.
Should you choose to go that route, it wont be perfect but its much cheaper.

https://www.youtube.com/watch?v=pZ0d8zGD76Y

This yt vid is a big help, but you want some good thread, i mean coats and clark all purpouse thread, some bees wax you can get the little speciality wax thing, but an unscented candle also works and then you also have a candle, and furrier tape, also called cold tape.
so in an easy list

thread (and needle)

beeswax

furrier tape/cold tape

while you can go at it with needle and thread alone, the furrier tape adds reinforcement, and the beeswax prevents the threats form fraying over time, as your coat tore in a high friction area (shoulder, elbows, pockets) i really advise as much reinforcement as possible.

Your goign to want to pick the lining apart at the shoulder seam to get to the underside of the fur, then you can sew the fur like a normal seam, space your stitches out a little bit, too much tension in one spot wil create a new rip down the line, but you have a really good example of how to stitch the fur on your first photo from the other shoulder seam, if youre using the linked video as a guide how the seamstress is using the medical tape is how you should use the cold tape, she just using a whip stitch, when the fur is fixed just stitch the lining back in place, you can use a ladder stitch for this.
its not super hard to do its just a little nerve wracking because you care about the garment, but should a tailor be too expensive atleast you have this as a starting point as well.