Correct hostname/PTR/A for a mail server serving multiple domains?

Apart from the purely technical answer (simple: chose whatever you want, consistency is sufficient) there is a lengthy list of soft considerations which all boil down to making it easy for you to maintain continuity (across rebrands/mergers/acquisitions and software/architectural modernisation), and making it easier for outsiders to quickly understand technical (and to some degree, legal) responsibility.

You have one mail server maintained by one entity serving the purposes of a group of companies all connected to my-Company. Express that:

• Make it obvious to outsiders what the server does, something that conveys "this is a properly maintained mail relay", like mx2 or mailout02
• If this server is not also serving an public incoming MX role, not being published as such is perfectly reasonable - connecting it to the domains as an authorized sender is done via SPF and DKIM
• Place it under a stable domain to minimize unusual permissions or ambiguities across different business entities, e.g. the "main" entity. If a wholly owned subsidiary tech controls services for a bunch of connected companies, that is reasonable. If the financial consultancy moneycity.example sends mail via the fishingclubcity.example domain that will rarely be visible to most non-technical staff, still complicate anti-phishing efforts.
• Combine that into one name and publish it, netting you an FQDN
• Setup that FQDN so it is used consistently. In the simplest case your mail server will greet (EHLO) with its one FQDN which resolves to the public addresses (one IPv4 & IPv6 each) used by the mail server which each also point (PTR/rDNS) point back to that one FQDN.

Correct hostname

I should note that especially the last bullet points is a recommendation which to require is arguably not correct as far as internet standards go, yet still practically highly desirable after a significant share of parties in the mail ecosystem have made it the easiest way to achieve their baseline requirements for some or all interaction with them.

Would I need an A record

Only when eventually netting A / AAAA records it becomes a resolvable name, which is required for use in SMTP context.

0.2.0.192.forgot-to-change-this.hosting.example I think it makes sense to change this

Yes. Not just because you might want to have full control of the domain where your hosting providers defaults might be unsuitable, but importantly also as a signal that you are declaring responsibility for the server, no longer judged by spam filters as an merely an average customer of that provider.

I won't serve a default site via nginx

Serving a redirect to your companies about-us or contact form page would not hurt, as long as it is not too complicated to maintain. Think of people failing to reach you for technical reasons looking for authentic alternates, by clicking on the server FQDN mentioned in the non-delivery returns.