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You can check this easily with Google Maps:
1. Driving from Wiesbaden to Kaiserslautern takes just over an hour (in off-peak and without jams) - while possible to commute daily, this is certainly not fun.
2. The fastest public transport connection from Wiesbaden to Giessen (centre to centre) takes 1 hour 50min, more depending on where you live and study in these cities. I would categorize this as not feasible on a daily basis.

Some trains from Wiesbaden to Giessen take 1.53 hours but these are the slow connections. There are also many connections that only take 1.31. All connections entail a change of trains. But one must also factor in the local transportation, getting to and from the stations in each town to where one really wants to go. Train connections can easily be found at the Deutsche Bahn site which is more reliable than the Google map routen planner train option:

I agree with Beppi that this seems excessive on a daily basis over a long period of time. Maybe if it is just once or twice a week (more likely for a student than a doing a job) and then just for a year or so one might manage it. I actually know people who are doing or have done such things. At least one can usually sit and read and study or work while in a train as opposed to driving.

But I also confirmed out of curiosity that Wiesbaden to Kaiserslautern by train is not easy. It usually takes over 2 hours and always entails 2 changes.

I think you’ll have to rethink this plan. Another option might be to live in either Kaiserslautern or Giessen. Then one of you would have to take a 2.30 to 3 hour, 2 change commute but then the other would be local to where they want to be.

Another consideration might be to live in Frankfurt. I personally hate Frankfurt but the train connections to Giessen are great, just 41 up to 57 minutes with no changes. And the connections Frankfurt to Kaiserslautern are 1.35 – 1.53 hours with 1 change – or 1.30 to 2 hours driving (bad!).

Another possibility could be living in Mainz. Like Wiesbaden it is a mid-sized, pretty, historical town. The train connections from there to Giessen are very comparable as from Wiesbaden. But the connection to Kaiserslautern if one ones to take train is better than from Wiesbaden; 1.26 – 1.47 with just 1 change. And for driving Mainz is a little bit closer on the same route to get to Kaiserslautern.