Usually, for Statement of Purpose (Ph.D. application), the standard length is 1000 words or 2 pages. However, the University of California, San Diego asked for a 2500 words limit. Does this mean I am expected to write closer to that length since my already written 1000-words length essay is less than half the word limit?
If you can write something useful in those 2500 words then yes. If you've written 1000 or 1500 words and you are 'filling' the rest to 2500 words by repeating stuff or being overly verbose, then no.
Do try to make it longer than 1000 words, they do give you all that extra room to elaborate so use it if you can. But when you really don't have anything else to say, you're simply done, whether it is at 1200 or 2400 words.
Yes, you should probably aim to get closer to their limit, I would imagine they have found 1,000-word SoPs to be insufficient, especially if that is some sort of standard in your field. I don't think you need to go all the way to 2,500 exactly, but less than half would probably not be a good look.