Will this shower set work as a replacement in my tub?
What you have purchased is a new shower valve that also comes with the "trim", a shower head and a tub spout.  Together, this is all commonly sold as a kit.  These are made for installing a new shower or a "down to the studs" remodel where you are going to cut the plumbing in the wall to remove the old shower valve and install a new one.

If this is what you intend on doing, then nothing has to match.  You are starting from scratch and you could use any shower valve.

Since you are concerned about "matching", what you probably wanted to purchase was a shower "trim kit" or "refresh kit".  These kits come with new handles and trim pieces but do not come with a valve.  There is no plumbing involved - you're just taking off the shower handle and putting on a new one (well I guess there is a little plumbing to install a new tub spout and shower head).

In this case, the kit must be made for your particular shower valve.  The control handles are the important parts because shower heads and tub spouts are universal (with a few exceptions).

Determining the brand of valve you have can be hard, but luckily you have a very well known valve known as "Mixet".  These were super common in the 80s (not totally sure on decade, but they were very popular).  With that brand name, you should be able to easily find a replacement handle or trim kit.