Do I have to move my domain registrar as well

No. Where a domain is registered and where its sites are hosted are separate arrangements, and moving hosting does not require moving the registration.

What actually has to change

Only where the domain points. That is a nameserver setting at your existing registrar, and it takes a minute to change in their control panel. The registration stays where it is, renews where it always did, and is billed by whoever bills it now.

Anyone telling you a hosting move requires a domain transfer is describing their own preference rather than a technical requirement.

Reasons people move registrations anyway

• Consolidation. Twenty domains across five registrars is five renewal calendars and five logins. Some people move for administrative sanity.
• A registrar they cannot work with. One that hides nameserver settings, charges to change them, or has no reliable support.
• Cost. Renewal prices vary considerably.

All are fine reasons. None of them are made urgent by a hosting move, which is why the two are worth keeping as separate decisions.

Reasons to deliberately keep them spread

On an estate there is an argument for not consolidating: registrations concentrated in one account are a single point of failure, both for an outage and for an account dispute. How far to take that is discussed in registrar spread and how far to take it.

If you do want to transfer

Do it separately from the hosting move, and afterwards rather than during. A transfer involves an authorisation code, an approval email, and a waiting period, and it occasionally resets nameserver settings to the new registrar's defaults.

Doing that in the same week as a hosting cutover means that if something breaks, you cannot tell which change caused it. Move the hosting, confirm everything works, then transfer the registration if you still want to.

The one thing to check before you start

Make sure you can actually reach the registrar account and change nameservers. Domains registered years ago by a former developer or agency are the classic blocker: the move is ready, and nobody can point the domain.

Find out who holds that access at the beginning rather than on cutover day. Everything else in a migration can be worked around; this one cannot. The wider sequence is in building a migration timeline.