"Is there really any law that makes it illegal to own copies of a publication?"
I couldn't find exact confirmation of which law Rahman was prosecuted under, but an article from 2012 describes it as "possessing an article for a terrorist purpose". A BBC article from 2007 uses the same phrase, with regard to "Section 57 of the Terrorism Act", which appears to refer to the Terrorism Act 2000:
57 Possession for terrorist purposes.
(1)A person commits an offence if he possesses an article in
circumstances which give rise to a reasonable suspicion that his
possession is for a purpose connected with the commission, preparation
or instigation of an act of terrorism.
(2)It is a defence for a person charged with an offence under this
section to prove that his possession of the article was not for a
purpose connected with the commission, preparation or instigation of
an act of terrorism.
(3)In proceedings for an offence under this section, if it is proved
that an article—
(a)was on any premises at the same time as the accused, or
(b)was on premises of which the accused was the occupier or which he
habitually used otherwise than as a member of the public,
the court may assume that the accused possessed the article, unless he
proves that he did not know of its presence on the premises or that he
had no control over it.
(4)A person guilty of an offence under this section shall be liable—
(a)on conviction on indictment, to imprisonment for a term not
exceeding 15 years,* to a fine or to both, or
(b)on summary conviction, to imprisonment for a term not exceeding six
months, to a fine not exceeding the statutory maximum or to both.
*Interestingly, the maximum term of imprisonment appears to have originally been 10 years, but was increased in 2006.
That law's not specific to publications, but in an attempt to answer your question more fully there does seem to be at least one law that "makes it illegal to own copies of a publication": the Criminal Justice and Immigration Act 2008, for example, renders possession of 'extreme pornography' illegal, and the definitions do not require (to reference your footnote) that the materials "imply an illegal act took place". Rather, the materials are illegal because (to quote a government consultation, via the Wikipedia page linked above)
such material may encourage or reinforce interest in violent and aberrant sexual activity to the detriment of society as a whole
That may or may not bother you from a freedom of speech point of view, but it leads me nicely onto a discussion of the underlying ethics/morality/rights debate that you impled in your question:
"We punish thoughtcrime, these days?"
As far as I understand, Rahman was in physical possession of the magazine. So no – not "thoughtcrime". Merely thinking about the magazine – or believing the views expressed in its contents – would not, of itself, be illegal. Indeed, as the article points out in its description of main case under discussion (i.e. the "Three Musketeers" case, not Rahman's prior conviction),
the prosecution described as the defendants’ “hateful beliefs” – but the judge, Mr Justice Globe, told them they could not convict based on “mindset evidence” alone.
However, like hate speech (speech, not thought), or conspiracy to commit a crime (which is what the men in the Three Musketeers case were being charged with), saying (or writing) certain things is, quite reasonably, illegal, if the intent of those words is incitement to commit a crime.
Restrictions on free speech exist everywhere, and any (political) debate around it generally centres less on an absolute right to free speech, and more on where the line should be drawn. Even in the US – which tends to draw the line at the permissive end of the spectrum – speech intended solely to cause damage is not immune from reproach. The classic example is "[falsely] shouting fire in a crowded theater":
The most stringent protection of free speech would not protect a man falsely shouting fire in a theater and causing a panic. [...] The question in every case is whether the words used are used in such circumstances and are of such a nature as to create a clear and present danger
Similarly, the publication of a magazine, the sole purpose of which is to (inspire or facilitate) damage, can be illegal. It would then seem reasonable for supplying that magazine to others to be illegal. Possession with a clear intent to supply (like with drugs) would also, reasonably, be illegal. And if drug possession, with no intent to supply, can be illegal, then why not possession of illegal material, even if there's no intent to supply?
I make no attempt here to justify (nor even to outline fully) the exact details of English law when it comes to these matters; however, there's an important distinction between freedom of speech (with regard to which nearly everyone accepts some restriction or regulation) and freedom of thought or conscience. To the best of my knowledge, there's no law (in Britain) that restricts either of those.