It helps registrars - who all know each other - work out roughly how many actual prospective pupils there are behind the total number of applications. Which helps them (a bit) in working out how many places to offer idc.
But no you don't have to answer, or answer fully, especially if you are still finalising your lost.
I don't think it makes any real difference to how likely it is you'll get an offer from a particular school.
Agree that it's for the school purposes to gauge likely numbers for entry.
They will look at where else you have applied and where you live,moles where any siblings go and make some assumptions - these will be for their predictions of numbers, not whether they offer you a place.
They will also in all liklihood ask your DC the same question, plus which is their top choice in interview. I would always tell my child to mention others (one or two) applied for but be a bit hazy about top choice unless they can honestly say it is where they are being interviewed that day.
Schools work with a formula based on previous take-up of offers. Some know they must make 3X the offers, compared to places, because they know they are not most people's top choice. Others know that typically 2/3 or 3/4 accept an offer. They can adjust their figures slightly from the previous year by knowing things like how many people have applied from so far away, they are unlikely to be serious candidates, how many have siblings elsewhere so might be less likely to come, how many have applied to much more aspirational schools so might not come if getting an offer there.
It's not a precise science and every year schools have to put on bulge classes or find they haven't got enough kids because they didn't get it right.
Don't worry about it too much. It's more for the glibal calculations than to affect if you get an interview or offer.
And yes, mentioning other schools just might make you more likely get a scholarship offer - if they think you've also applied to a compeitor and are likely to get an offer from them....and if you've performed well in the exam...they might then be more likely to give a scholarship offer if you are a marginal scholarship applicant.
Honestly, they just need to know how many will have date clashes with other schools. Some schools ask where siblings go to school and might be more tempted to take your DC if there is another in the pipeline but most just want to plan their extra sitting.