Photo- what is this called and does it matter if its broken?

looks like a dust boot for the strut. unless they have been replaced, it might be just as easy to do them both at the same time.

 

The shock itself looks to be in good shape. There was a thread on this subject not long back, and I think the concensus was to patch it back together, say with some small zip-tie "stiches", then some tape over to make a bit more secure and help it in it's function, which is as a dust shield.

Another option: there might be aftermarket split boots that you can put on? Not sure.

It's a big/expensive hassle to replace the boots, and I think most who do it replace the shocks at the same time?