If you want something slightly longer than a headline for the explanation why the left/progressives call this a ban:
The Texas GOP passed a law that effectively bans abortions after the sixth week of pregnancy — before many women even realize they are pregnant — but outsourced enforcement of this ban to private citizens.
Of course, someone in the Saudi rulership might say that abortion is not banned in their country... if by that they mean that it's still allowed in order to save the life of the mother, but otherwise rather banned. (Ok, they prefer the word "forbidden" when it comes to Islamic countries, but they use "ban" when talking about non-Islamic ones.) See also Poland, which passed a similar law (to Saudi one, I mean). The BBC headlined "Poland enforces controversial near-total abortion ban".
Also the Texas ban is not as ineffective as someone suggest... WaPo quoted:
This week, John Seago, the legislative director for Texas Right to Life, told NPR, “We have a network of pro-life attorneys and pro-life activists who even now give us tips and send us information that may lead one to believe that the law is being broken by the abortion industry.”
According to WaPo the conservative strategy is to use such organizations... pretty much mirroring how ACLU etc. bring lawsuits "on the other side" for civil rights etc. Also WaPo mentions:
The law also bars courts from awarding “costs or attorney’s fees … to a defendant in an action brought under this” law. Even if abortion providers win the lawsuits, in other words, they will incur costs — financial, emotional and otherwise — while their opponents’ costs and potential losses are limited.
So if you want to be very charitable with the conservatives' viewpoint, you could say that the law [at least] gives the right to impose private fines on abortion providers, limited in monetary amount only by the number of people who are willing to sue a given provider. Fox News uses the term "outlawing" instead of ban... even that sounds pretty awkward in context (really "a law outlawing abortions"):
Texas implemented a law this week outlawing abortions once medical professionals can detect a fetal heartbeat, usually around six weeks into a pregnancy and before many women know they are pregnant.
and they also have coverage of the first lawsuit already launched under this new law.
(Frankly, the Q would have been more suitable on politics SE than here, as an ACLU news release is not a legal document, so it's not using strict legal terminology.)
If you're curious, Black's fabled Law Dictionary, defines ban rather expansively (4th ed. quoted here)
In Old English and Civil Law:
A proclamation; a public notice; the announcement of an intended marriage. [Cowell.] An excommunication; a curse, publicly pronounced. A
proclamation of silence made by a crier in court
before the meeting of champions in combat.
[Cowell.] A statute, edict, or command; a fine, or
penalty. An expanse; an extent of space or territory; a space inclosed within certain limits; the
limits or bounds themselves. Spelman. An open
field; the outskirts of a village. A privileged
space or territory around a town, monastery, or
French Law: The right of announcing the time of mowing,
reaping, and gathering the vintage, exercised by
certain seignorial lords. [Guyot, Repert. Univ.]
Old European Law: A military standard; a thing unfurled, a banner.
Spelman. A summoning to a standard; a calling
out of a military force; the force itself so summoned; a national army levied by proclamation.
Note that I'm not claiming this is the sense in which the ACLU is using the term, rather than the modern synonym "to prohibit".
Also, if you peruse almost any medical or human rights article on the matter they freely use the term ban in the same way as ACLU does, e.g.
The impact of abortion bans on women’s health in the Middle East and North Africa (MENA) region is understudied [...] In contrast to extremely restrictive countries such as El Salvador and Poland, and, more recently, several US states that ban nearly all abortions, all countries in the MENA region permit abortion if the pregnant woman’s life is in danger. [...] In most cases, despite the social and legal ban on abortion in Lebanon, women are able to terminate one or several pregnancies under medical supervision in private hospitals and doctors’ offices.
So it really depends in which circles you're swimming whether this is a proper usage of the word or not.
FWTW, AMA's reaction/headline is
Texas SB 8 puts bounties on doctors’ heads for delivering care
They got the legal angle somewhat better as to what SB.8 actually does, but then there's of course the choice of language how to describe that...