I'm torn on this question; most of my life I just had a tenor. There is something to be said for playing the same horn all the time. Once I bought a cheap horn in nice shape when I had to send my main one off for an overhaul, then sold it when I got my horn back. Later on I finally got a nice alto but still had just the one tenor. During this time I playing a lot too, but the six-nighters were a thing of the past. That really ages a sax. Next came my first 'back-up' tenor, a Selmer USA. The original neck was in great condition but was awful - if I didn't have a replacement MK VI neck to try, I would not have been able to use the horn - but fortunately, that neck was very good on it. I soon got a baritone too, then a soprano.
So the real question is whether or not you 'need' a back-up for your principle instrument. I'd say that depends entirely on how much you play, and whether you always work from home or you tour.
I find myself now with a back-up that I'd rather play than my #1. The good thing about that is, I'll sooner or later swing the other way and play the VI for a few months. I do use the same mouthpiece on both. The bad thing about this is every horn is different - I know its true that if you stick to the same horn you will usually be better/more accurate, but in my situation of nearing the end of my playing time, I'm going for the fun. I don't care about the future or my 'career'. I just want to play whichever tenor suits my fancy on a given night. As far as having all the other saxes, playing them does not seem to affect my tenor playing even though mostly the mouthpieces are wildly different from the tenor and from each other. I said in another thread that I believe you get used to a certain mouthpiece on each sax and this does not cross you up going directly from horn to horn. For example, I played Level Air on tenor, alto and bari for many years. After I changed to Guardala on tenor and alto in 1989, after I started using it on tenor I could no longer play the Level Air. I mean I could make it play but it was just too 'foreign' - after using it for 23 years! Now here's the strange part - I can happily play Level Air on baritone and go directly back and forth with no problem!
Anyway, there are different ways to approach the back-up horn, and I think most guys are like me - we end up with two or more principle saxes and they both/all play great. The other way, which is probably not worth the trouble, is to have an 'emergency' principle sax that you really don't like that much but will get you through a gig. I think most of us will not keep a horn that 'won't blow' and/or just doesn't cut it on your gig - it's pointless.