Is there a way to have the same season in travel locations as in the homeworld? And for the homeworld to progress in seasons while travelling.

No.

Sorry, that was cheeky of me. I must be in one of my moods today.

First of all, no because homeworlds do not progress at all when they are not being played actively. They are supposed to remain in suspended animation while your active sims are elsewhere, and time does not move again until they return.

But I do know what you mean on the destinations' seasons not matching up. That can be annoying, but remember that in the real world not all of us are on the same season at the same time anyway (Northern vs. Southern Hemispheres). What we can do is just use the testingcheats shift-click on the ground trick to change the season at our destination to what we think it should be upon arrival, if that's any help.

Cheeky is cute I didn't think seasons would change while travelling. My sim is always in summer. Thanks for the TCE trick, that would work.

Ah yes, and the other reason this would fall apart if sync were to be attempted is that there is no reason to assume that all connected worlds have the same season lengths or even the same sets of seasons allowed. Mine are all different from each other. But yes, all summer all of the time for frequent travelers would be highly annoying, I'm not sure that was intended even by EA so thankfully we do have the TCE solution.

The length of the homeworld seasons is roughly 8 days. Now she's in France, the seasons have changed to 15 days for summer, an old setting I had before.