Consider a sentence:
They behave like us
Why do we use "us" here? Should it be subjective pronoun "we" as it is a subject of implied verb "do", i.e.
They behave like we (do)
Which one is correct and why?
You've got it backwards. First, you look at what people actually say, then you come up with an analysis. If people actually said *they behave like we, then the ellipsis analysis would be reasonable. But since we don't say that, the ellipsis analysis must be wrong.
Instead, like must be introducing a noun phrase, not a verb phrase, and so the accusative us is the natural choice. This is perfectly standard and has been for hundreds of years. Of course, it can introduce a verb phrase as well:
1. They behave like us. (standard, introducing noun phrase)
2. They behave like we do. (standard, introducing verb phrase)
Historically, some prescriptivists have taught people that option 1 is incorrect, despite the fact that it's standard English. But even those who reject option 1 above would not omit do:
3. *They behave like we. (non-standard omission of do)
So if you want to say we, make sure you say do as well.
In this answer, * marks a sentence that is not standard English.