The following resonance structures are possible for the given compound. (a) promotes both aromaticity and Anti-aromaticity whereas (b) promotes aromaticity. In the resonance hybrid, (b) contributes more as we can see. Will the resonance hybrid be called aromatic or anti-aromatic?
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From a classification point of view, there is no reason for not calling the molecule aromatic.
First, saying a molecule is aromatic doesn't imply that aromaticity fully extends over it. A trivial example is, just to say, cyclohexylbenzene, or whatever aromatic molecule of your choice.
Now, let us assume the two rings being completely independent so to exclude any delocalisation. The phenyl moiety will be just aromatic while the Huckel antiaromatic cyclobutadiene ring will simply adopt a rectangular shape, so effectively avoiding the antiaromatic unstable configuration.
Again, there wouldn't be reason for not calling the molecule aromatic.
This is often a sort of misunderstanding when antiaromaticity is discussed. It is just a criterion to decide if full delocalisation within planar rings is attained. If the answer is no, because the system is in fact antiaromatic, then there is nothing actually specific to attain.
In short: while a ring that is Huckel-aromatic does actually tends to be aromatic, the Huckel-antiaromaticity is simply a relative high energy situation that a system will always try to avoid.*
Finally let us consider the actual case. The resonance structures you wrote indicate indeed a certain character of aromaticity in the cyclobutadiene moiety and a partial double character of the interring bond. This alone, let aside the above considerations, definitely tell us that the molecule is aromatic.
For more insights I've found the following papers, analyzing delocalisation in phenylcyclobutadiene experimentally (enthalpy of hydrogenation and EPR) and theoretically.
As a very quick resume enthalpy of hydrogenation point to a destabilising effect of the Huckel-antiaromatic moiety, while electron spin resonance and calculation show overall aromatic rings current and an interring C-C bond length of about 1.44 Å, about midway from a single to a double C-C bond length.
*note that chemists are sometimes able to constrain structures in otherwise highly unfavourable systems. I won't be surprised if exotic beasts forcing an inner ring into antiaromaticity were synthesised. I might even have seen a related Q here in SE.