"I take on board your thoughts" in a formal setting

Solution 1:

'Take on board' does have conflicting senses:

take on board

To acknowledge and consider implementation of or action on a piece of information or a viewpoint, often one that has been newly presented.

[Farlex Dictionary of Idioms]

take on board

To accept a new idea or piece of advice, and act accordingly.

[topmeaning.com]

So sometimes used to include implementation of recommendations, sometimes used only to include consideration of recommendations.

In OP's example, the only logical reading is that the speaker is saying that they feel your thoughts on X require at least serious consideration (while they're with you all the way, including implementation, on Y). Of course, this covers only the language, not whether people change their minds etc.

Solution 2:

You are correct in having misgivings.

https://www.expatica.com/uk/working/employment-basics/dutch-british-business-102530/

What the British say: I hear what you say. ("I take on board your thoughts on X")
What the British mean: l disagree and do not want to discuss it any further.
What the non-British understand: they accept my point of view.

That said, your partner may indeed mean what they said. I would perhaps double check.