The claim that socialism doesn't allow for private property ownership is mostly true because socialism generally rejects private ownership of the means of production, though it typically distinguishes this from personal property, which is often allowed.
Socialism doesn't allow for private property ownership.

Who Owns Property in Socialism? — Jeffrey Stamberger
Source: jeffreystamberger.com
Key Evidence
Socialism rejects private ownership of means of production but allows personal property.
iep.utm.eduiep.utm.eduSocialism
reddit.comreddit.comr/Socialism_101 on Reddit: socialism view on private property
Private property in socialism refers mainly to productive assets like factories and farms.
reddit.comreddit.comr/Socialism_101 on Reddit: If Private Property doesn't exist, does Socialism...
reddit.comreddit.comr/CapitalismVSocialism on Reddit: For Socialists, do you believe in private...
Some socialist variants permit private property rights outside production means.
quora.comquora.comCan socialism coexist with private property rights? Is there a specific type of...
quora.comquora.comIs there a form of socialism in which you can own private property?
What the Evidence Shows
Socialism as an ideology primarily opposes private ownership of the means of production, such as factories, farms, and large-scale capital assets, because these enable wealth accumulation by a few at the expense of the majority. This is a core principle reflected in multiple sources.iep.utm.eduiep.utm.eduSocialism
link.springer.comlink.springer.comProperty Rights in Socialism, Exchange and Production+2 more
However, socialism usually differentiates between 'private property' in the economic sense (productive assets) and 'personal property' (items for personal use like clothing, homes, or personal belongings). Most socialist theories and movements accept personal property rights while advocating for collective or public ownership of productive resources.reddit.comreddit.comr/CapitalismVSocialism on Reddit: For Socialists, do you believe in private... There are also variations within socialism; some forms allow limited private property rights outside the means of production.
quora.comquora.comCan socialism coexist with private property rights? Is there a specific type of...+3 more
Historical socialist experiments have focused on abolishing private property rights related to production but not necessarily all forms of private ownership. Therefore, the statement is accurate in the context of economic/private property but incomplete if interpreted as banning all forms of private ownership.