Dressed Up Just Like Like The Union Jack?

I recently heard the Rolling Stones song "Get off of My Cloud" and although it wasn't the first time I have listened to the song, some of the lyrics jumped out at me:

Then in flies a guy
Who's all dressed up just like the Union Jack
And says I've won five pounds
If I have his kind of detergent pack?

I don't know of many people that were dressed up during that time in a Union Jack coat other than Pete Townshend. So was Mick Jagger singing about Pete? Hard to know really. Wikipedia doesn't mention anything about it. I do know that The Who and Rolling Stones were playing the same scene in London around the time the song was written (1965). Pete was and is a huge Rolling Stones' fan. But the "detergent pack" reference would make more sense if the song came out around the time of The Who Sell Out album in 1967. It would be a worthwhile question - at least to me - for Pete and/or Jagger.