Asking prospective FILs permission

If his future fiancée were opposed to such a tradition he would know about it.
Some women can find offensive.
However if he thinks she'd be fine with the gesture then I'd go ahead with it.
But frame it as pp said.
The FIL can't say no.

DS1 rang California to ask his DFiL. He said the worst thing was that he wasn't home so he had all the build up a second time to speak to him. He found it nerve wracking even though they knew and liked each other. It was very much appreciated.

One SIL did ask me in the absence of DD's Father. He did more as a statement of fact rather than a question.
SPF

I don't think it's necessary, but perhaps it's polite and like others have said will give him brownie points. 12 years ago I was asked the same question by my son. My advice was perhaps he should say " I'm going to ask daughter to marry me and I hope that you approve" worked a treat. Good luck to your son and congratulations to the happy couple.

Yes you are right Jalima, its a nice thing to do and makes for good future relationships.

Son in Law ( not Silver ) ?

Yes. DS spoke to his prospective FiL as his 2 BiL's did before him. FiL was delighted and all 3 boys started off on the right foot with the in-laws who hosted 3 beautiful weddings.
FiL was so pleased to be included in rugby outings and thoroughly enjoyed the stag do of each groom.
Recently DD1's boyfriend made the token gesture even though he's 45 and she's 41!

My now SiLver drove from Bath to N London to ask for our daughter's hand. Albeit they were both in their 40s. So a lovely gesture. And SiLver has turned out to be a very special person.

We thought it was quite quaint when DH asked my father for my Hand. Actually he was very relieved to get rid of me

I think it is one of those old fashioned customs which has been resurrected by the young. I have heard of several such recently, including future SiL who asked us if we would approve of him proposing. We were delighted! As Ginny has said, the hard part was keeping quiet and not spoiling the surprise.

Yes, SIL asked DH although he and DD had already talked about it. DH was pleased that he 'asked'.
DS and DDIL's father went for a chat over a pint at the local pub, a bit of future FIL/SIL bonding.

Even if it's not necessary, it is a nice thing to do and good manners.

Not necessary but a nice thing to do. Our SIL spoke to us together . The hardest part was knowing when the proposal would happen and not letting on to our DD.

It might be old fashioned but he will get an awful lot of brownie points if he does ask her father! My DH was very impressed that son-in-law asked him - and actually mentioned how pleased he was to be asked when he did his Father-of-the-bride speech at their wedding recently.