Yes, but only the messages on your device. Once you run out of room on your phone, switch to another phone, use your laptop, etc, all that history is gone forever, and obviously not searchable.

Element actually does encrypted search a bit better than Signal. Signal stores every message it receives, in its entirety, forever (unless you tell it not to, or course, but then you can't search). That makes search dead simple, but it also fills up my phone with hundreds of megs a month. Element stores messages on the server, so old ones don't need to stay on your phone (only your keys). But you can still search the entire history because the client indexes every message that comes in and stores that index forever, which is much smaller than a bunch of media files, and even the plain text.