1:1 with Engineering Manager during Internship

Great questions!

I am wondering if it is necessary to also have a weekly meeting with the engineering manager as well.

I think it has value but it definitely shouldn't be weekly. If you have an intern -> intern manager -> manager model, the manager is generally very hands-off with the intern (that's how it worked at Meta). You wouldn't have nearly enough to talk about in a manager 1 on 1. I am thinking monthly cadence.

When it comes to 1 on 1 topics, I need to make a course about 1 on 1 meetings (my course backlog is so big 😅), but in the meantime, I recommend:

1. Small talk - Genuinely learn about each other's lives. If you play your cards right, that manager can be a strong vouch for you long-term, well after the internship.
2. Feedback - This is the big one, but it's tricky with the higher-level manager. Regardless, I still think it's worth asking. The value-add from talking to the manager is they'll have a more holistic view, aggregating feedback about you from all members of the team.
3. Longer-term advice and logistics - This is the main value add from the manager as it's their job to think longer-term. One more immediate "longer-term" thing to talk about (especially if you're early into the internship) is how the return offer system works.

I am not a huge fan of project updates as that's very tactical and that alignment should come naturally from Slack/Teams messages, your intern manager/other peers bubbling things to the manager, and any status updates you publish (emails, Workplace posts if you're at Meta, bigger Slack/Teams messages). Even with your intern manager, this isn't a good topic - It will be even worse with the manager. 1 on 1 meetings are more about the awkward stuff.