The most significant aspect of "male socialisation" is coldness from strangers beginning at a very young age
Internet discourse is way too keen on pretending "male socialisation" occurs primarily through "education". IE, boys are "taught" to be violent, aggressive, domineering etc. This is a naive view about what "socialisation" entails. People do not need to be consciously deciding to treat you different in order for their actions to influence how you interpret the world; this is true of everyone.
I would argue a predominating factor in male socialisation is that there is a degree of coldness with which boys are treated that increases as they age, and especially after puberty. This does not fall on all boys evenly, and of course there are social butterflies who are gregarious from a young age, but the outcome of this tendency towards coldness socialises boys into being the "first movers" of social interactions, if they are interested in obtaining their social needs (which differ for each individual). Some come to this naturally, some grow to resent being forced into this, and others simply blend into the walls and pass through social interactions not particular standing out. This coldness comes from both men and women, children and adults, and is a massive factor in gendered socialisation that is easily ignored because we each only have our very particular social experiences to go on.
"Coldness" is an important concept because it's not "cruelty". There is often no malice behind it, and it is performed totally unconsciously. However, it slightly conditions every social interaction you have from quite a young age, and impacts your social development in a way you a likely not conscious of. Subconscious bias theory has been somewhat mainstream for years at this point, but "gender socialisation" is still thought of in terms of how we "teach" children vs how we already "treat" them as members of their gender, and the expectations that accrues.
I won't speculate much further, but I think many gendered differences are downstream of this "coldness", which in effect leads to boys "competing" over attention in ways slightly different than girls do. These differences don't need to be large in order for them to compound throughout our lives.