I really like the intuitive profile page of Meta Stack Exchange. Is there a way to upgrade the same in a different community, for example, Ask Patents?
You don't have to request this. It will rollout eventually to all sites.
The process is explained in New profile pages are out - bugs and feedback master list but if your site can have the new profile depends on if the new CSS framework is rolled out as indicated by Oded:
The new design is dependent on a SE network-wide update (a new base css framework) that the design team have been working on. This is an ongoing effort and we will be launching the new pages as sites are moved to the new design.
The list of sites and it's progress can be tracked here.
Ask Patents is the last in the list On Deck but I'm not sure if that means you'll come last. The list is empty, all sites that were eligible are now updated. Give it another 6-8 weeks. Took only a little bit over 6 to 8 months.