Remember that every technical product is the result of a lot of compromises, or with a better word design decisions.
The engineer designing the antenna will work according to a specification that will outline the priorities for the design. It will include things like: frequency range, amplification, directivity, max ratings for voltage and power, price, size, SVR, weight, materials, temperature range, longevity, ...
This means that a very high amplification very directive antenna with a huge budget (think radio telescope) will look very different from a 5 cent radio transistor antenna.
Trying to use the antenna outside of, say the intended frequency range, then you are on your own. It might work very well, it probably will not be great, but it might be acceptable. The exact design of the antenna might give an experienced person clues to the performance but the proof is in the actual performance.
Transmitting a 40 MHz signal using a 868 MHz antenna will probably result in very poor amplification, meaning that most of the energy will be transformed to heat and very little inte actual radio waves. It might as well show very odd directionality perhaps sending most of then radio energy into only a small lobe. Same effect might happen with a 868 MHz antenna on a 2.4 GHz router -- most of the transmitter radio signal will turn into heat (or perhaps not, depends on the exact design of the antenna). And similarily for the receiving end, the antenna might be quite "deaf" at frequencies outside the intended range, or not. Mty guess is that it will not give a satisfying result.
Additionally, it might hurt the transmitter. Pumping energy inte a circuit and not letting it exit the intended way through the antenna may hurt components and you will see the magic smoke exiting from them. Admittedly, in low power situations and/or consumer electronics the circuits are probably protected so harm may not occur.