Cached forms can have duplicate HTML IDs, which disrupts accessible form labels

I don't really know if this is a core issue or block cache alter issue... even I don't know if it there is any solution...
I have a website with both "User login" block and "Search" block, cached with "Globally cached" option. But when using cache on that blocks, I get duplicated ID's (i.e., #edit-submit and #edit-actions). I think it's because admins don't see "User login" block, so "Search" block HTML gets cached with #edit-submit ID. Then, when first anonymous visitor comes to the site, "Search" block HTML gets loaded from cache, and "User login" block it's dinamically generated with #edit-submit ID (because it wasn't generated dinamically before). So, the HTML generated can't be validated by automatic tools.
This also can generate bugs in javascript, and stylesheet problems (luckily for me, it's not my case).
Is there any way to avoid this? Storing somewhere IDs cached used when caching or something...
As I said, I don't know if this is a core issue or not.