Hakone Free Pass

Is it allowed if I use Hakone Free Pass to travel from Tokyo to Hakone and return on the same day?

From Shinjuku on the Odakyu railway, yes. The version sold at Odakyu Shinjuku includes one round trip.
The "Romance Car" is an extra/supplement and optional.
Many people use the pass for a day trip to Hakone if doing the "loop".
Note: With Owakudani off limits, the ropeway is not operating, so the FreePass may be more expensive for your visit as you cannot use it on that particular leg of the loop. Certainly when I visited from Odawara it would have been more expensive when I did the lop with guests back in May.

That's what I did, and many others. Although there really is enough there for two days. And not every place there is crammed with tourists. I went to the Botanical Garden of Wetlands and it was nearly empty, except for a few couples and a cat. And I had the Choanji Temple all to myself - it was magical.