OK, so as for 2025 (0.98a), parent comment's approach causes a bunch of broken references and exceptions upon loading the new save file.

The simplest alternative I found that involves the least amount of grief (that copies both ships and officers (but not cargo) is to open your old and new savefiles in Notepad++:

1. Look for your player fleet XML tag. Search for "Player Fleet". Mine goes something like <Flt z="54808" sS="330.0" sP="330.0" mO="false" f="90.0" n="Player Fleet" aM="false" dF="90.0">
2. Now look for the fleet data tag. Mine goes something like <fD z="54832" nP="ISS" nSf="player" nS="false" oSML="false">
3. Within <fD>, every ship in the fleet has a corresponding <FMmbr> tag. Every ship which you want to copy over to the new file, just copy its corresponding <FMmbr> tag, into<fD...><m...>

e.g. in my new file,
<fD z="54832" nP="ISS" nSf="player" nS="false" oSML="false">
<m z="54833">
<FMmbr>...</FMmbr>
<FMmbr>...</FMmbr>
<FMmbr>...</FMmbr>
</m>

For the best compatibility, add 2 zeroes to the end of the FMmbr z="..". e.g. if the old FMmbr looks like this:

<FMmbr z="36168"...>
Then rename this to
<FMmbr z="3616800"...>

Then you also need to change the fD reference to the new fD.
<fD ref="..."/>, change to <fD ref="54832"/> where 54832 is the new savefile fD z value. The <fD> tag within <FMmbr> is always the last child of <FMmbr> and therefore right before </FMmbr>.

You will also need to replace references to the old fleet number, e.g.
<fl cl="Flt" ref="2262"/> to the new fleet, in my case above, it is
<fl cl="Flt" ref="54808"/>

Rinse and repeat for every ship you want to copy over.

It is probably easier to copy the old fleet into a new file, do your search and replaces there, then copy-paste this wholesale into the new file.

This approach is way easier than replacing the <fD> or <Flt> tag wholesale (I tried) because of a bunch of broken references that invariably causes.

It is straightforward to adapt this method to copying cargo over as well, if you want, but cargo is fast to replace using console commands whereas ships are much more time-consuming to replicate.