Do the UK or the Italian government keep a registry of entries/exits for EU citizens?

They check your passport to see if you are eligible to enter their country.
They often also check if you are on some wanted list

• for EU Citizens, entering or leaving the Schengen area, nothing more is done

Depending on the jurasdiction and citizenship, the enter/exit information may be stored.

Starting the 1st of January 2021, EU Citizens should assume that their entry into the United Kingdom will be noted and a general leave to enter will be granted up to 6 months for short term visits. The advice is being given to use a Passport rather than an ID.

Starting the 1st of October 2021, for short term visits, a Passport will be required for EU Citizens (with the exception of Irish citizens).

What you need to enter the UK
If you’re an EU, EEA or Swiss citizen
...
You will not be able to use your EEA or Swiss national ID card to enter the UK from 1 October 2021.

You can continue to use your national ID card to enter the UK until at least 31 December 2025 if you:

• have settled or pre-settled status under the EU Settlement Scheme

...

Bring your passport rather than an ID card
If you are an EEA citizen, you may find it faster to use your passport rather than your EEA national identity card at the UK border.

This is because:

• passports are faster for our Border Force officers to process
• you can use your EU passport at our eGates

Sources:

• The Border Operating Model - GOV.UK
  • version 2020-10-08, Page 259 of PDF
• Visiting the UK: information for EU, EEA and Swiss citizens - GOV.UK
• Guide to faster travel through the UK border - GOV.UK

Italy doesn't record entry/exit for EU/EEA citizens.

Italy is member of the Schengen Area, so their rule apply first.

According to this Wikipedia page, recording the entry and exit is not required. GDPR, in particular, requires that all personal data processing are done only when they are required by a need or regulation.

If you are a EU/EEA citizen, which includes Italy, there is no need to record about your travel.

Please do not confuse this with the systematic passport control even for the EU/EEA citizens. Border Police needs to scan your passport to ensure it is not stolen and there is no arrest warrant on your name both on Local and European databases.

The UK most certainly does, as you can request the past five years of your own travel records:

From: https://www.gov.uk/government/publications/requests-for-personal-data-uk-visas-and-immigration/request-personal-information-held-by-uk-visas-and-immigration

Basic

You can request a copy of:

• an electronic summary of your immigration history
• landing cards we hold electronically
• visa applications you submitted from outside of the UK (as part of entry clearance)
• Workers Registration Scheme (WRS) information if you are a national of Czechia, Estonia, Hungary, Latvia, Lithuania, Poland, Slovakia and Slovenia
• Entry and exit into and out of the UK. Your travel history can be provided for the past 5 years if a passport or travel document is submitted for this period

The last point seems to match perfectly your case.