I mean, let’s take rabbinic Judaism and Christianity. Both of them stem from the 2nd Temple Judaism, Christianity has the Tanakh as part of its Scriptures.

If a Christian claims that his God is the God of Judaism, it’s a consistent view rooted in history.
If a Jewish person claims that his God is not the God of Christianity, because the latter is Jesus, this is also a consistent view, rooted in logic.
There shouldn’t be a single “objective” view that the God of Judaism and Christianity is or isn’t the “same” one. It will always depend on the believer.