It’s traditional in my family. We’re Ashkenazi Jewish.
The way it works (in my family, I’m not saying the way other families do it is wrong) is that the religious Hebrew name stays the same, but the English name changes and is often/usually initial-matched.
It makes more sense with examples. Okay, my grandfather was a [name]Harvey[/name] in English; from [name]Hanan[/name] in Hebrew. HIS granddad was also a [name]Hanan[/name]. His granddad, however, was NOT named [name]Harvey[/name], that was a “You’re in [name]America[/name], you need an English name!” thing. Okay so far?
So if I name after my granddad, which I very much want to do, the normal thing in my family would be to make his religious (like baptismal, I guess?) name [name]Hanan[/name] for use in services, at his [name]Bar[/name] Mitzvah, etc. Then his English, legal name any dang H name I feel like. It would be considered unnecessary to match the [name]Harvey[/name] as the English name doesn’t really matter all that much from a religious standpoint, it’s just a nice connection. My granddad [name]Harvey[/name] would himself be confused if he knew I was picking a wildly out of style English name.
Does that make sense?
In my sub-culture (Sephardic Jews do it differently…) we name after dead ancestors. But never live ones! So I would not - and hopefully never have to consider - naming after my parents, as they are alive and hopefully will be for a long time. But my grandparents are dead and I really want to name after them. We’d use either their first or middle initial.
Like my grandmother - sorry for making this about my family but its easier with examples - was, in English, [name]Irene[/name]. Her Hebrew name is Chaia, which is very difficult to use as an English name because it sounds like “Hiya!”
I am thinking of naming after her, [name]Ivy[/name]. My cousin already named after her, [name]Ilana[/name]. [name]Both[/name] great-granddaughters would be considered to be named after [name]Irene[/name].
Does that make sense?