It is perfectly acceptable to apply to more than one school and many schools suggest you do because their
school is so oversubscribed!
It's not uncommon for St Paul's, Westminster and CLB to be registered or Westminster and Eton or Eton and Harrow, or Eton and Winchester. The first example is keeping options open, the second is a boarding option, the third is social climbing as most OH or OE's would never apply to the rival school.
My son sat for Eton, Westminster and Winchester and it had no effect, nor did the schools contact one another. I think the major schools are far too busy and have too many applicants to contact the other schools nowadays although it may still happen.
Some schools have earlier pre-tests which works out well as you have a decision before having to apply to another school. Unfortunately E, W and W do not but Harrow's test is a year after these ones.