What is the current stance of the Catholic Church regarding the legality of non-Catholic religions?

Although, of course, the Catholic Church regards itself as the one true Church, the Second Vatican Council has fully reversed policies put in place by Pope Pius X, when he attempted to stamp out all traces of 'Modernism'. The Council calls on all Christians to respect as a human right, given by divine revelation, that others may worship both privately and publicly as they choose, and calls on all governments to allow freedom of worship without coercion. It was able to do this because Pope Pius IX did not speak ex cathedra when he issued his Syllabus of Errors.

Among other things, the council's 'Declaration on Religious Freedom', which bears the subtitle 'On the Right of the Person and of Communities to Social and Civil Freedom in Religious Matters' states:

2 This Vatican Council declares that the human person has a right to religious freedom. This freedom means that all men are to be immune from coercion on the part of individuals or of social groups and of any human power, in such wise that no one is to be forced to act in a manner contrary to his own beliefs, whether privately or publicly, whether alone or in association with others, within due limits.

3 ... Truth, however, is to be sought after in a manner proper to the dignity of the human person and his social nature. The inquiry is to be free, carried on with the aid of teaching or instruction, communication and dialogue, in the course of which men explain to one another the truth they have discovered, or think they have discovered, in order thus to assist one another in the quest for truth ... In all his activity a man is bound to follow his conscience in order that he may come to God, the end and purpose of life. It follows that he is not to be forced to act in a manner contrary to his conscience. Nor, on the other hand, is he to be restrained from acting in accordance with his conscience, especially in matters religious.

4 ... Religious communities also have the right not to be hindered, either by legal measures or by administrative action on the part of government, in the selection, training, appointment, and transferral of their own ministers, in communicating with religious authorities and communities abroad, in erecting buildings for religious purposes, and in the acquisition and use of suitable funds or properties. Religious communities also have the right not to be hindered in their public teaching and witness to their faith, whether by the spoken or by the written word.

9 The declaration of this Vatican Council on the right of man to religious freedom has its foundation in the dignity of the person, whose exigencies have come to be are fully known to human reason through centuries of experience. What is more, this doctrine of freedom has roots in divine revelation, and for this reason Christians are bound to respect it all the more conscientiously.

Pope St John Paul II and Pope Francis were acting in accordance with the directives of the Vatican Council when they participated in interfaith prayers for peace alongside not only non-Catholic Christians, but also leaders of other religions.

it would seem that Pius IX orders that all religions bar Catholicism are to be declared illegal by the State.

States are to prevent false religions from publicly spreading their errors, but the false sects can be tolerated. Unlike Islam, Catholicism does not permit States to force-convert its citizens to Catholicism.

In fact, in medieval Christendom when the State recognized the Church as the true religion, Jews were permitted to raise their children in Judaism as long as they did not spread their errors and heresies publicly in society. It is also forbidden to baptize children of unbelievers against their will.

The main point of Pope Pius IX's statements is that States are able to recognize Catholicism as the true religion and are obliged, for the sake of the common good, to suppress false sects and prevent them from spreading their dangerous errors and heresies.

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The contradiction you mention between pre- and post-Vatican II teachings on Church-State relations is one reason the Society of St. Pius X (SSPX) and other groups think that Vatican II taught error and was therefore an invalid council. For example, read ch. 11 "Religious Liberty" of the SSPX's founder Abp. Marcel Lefebvre's Open Letter to Confused Catholics.

Compare what I number below in the Vatican II document Dignitatis Humanæ (7 Dec. 1965):

2. This Vatican Council declares that the human person has a right to religious freedom [1]. This freedom means that all men are to be immune from coercion on the part of individuals or of social groups and of any human power, in such wise that no one is to be forced to act in a manner contrary to his own beliefs, whether privately or publicly [2], whether alone or in association with others, within due limits [3]. … This right of the human person to religious freedom is to be recognized in the constitutional law whereby society is governed and thus it is to become a civil right [2].

to my corresponding numbers in Pope Pius IX's encyclical Quanta Cura:

For you well know, venerable brethren, that at this time men are found not a few who, applying to civil society the impious and absurd principle of "naturalism," as they call it, dare to teach that "the best constitution of public society and (also) civil progress altogether require that human society be conducted and governed without regard being had to religion any more than if it did not exist; or, at least, without any distinction being made between the true religion and false ones." And, against the doctrine of Scripture, of the Church, and of the Holy Fathers, they do not hesitate to assert that "that is the best condition of civil society, in which no duty is recognized, as attached to the civil power, of restraining by enacted penalties, offenders against the Catholic religion, except so far as public peace may require [3]." From which totally false idea of social government they do not fear to foster that erroneous opinion, most fatal in its effects on the Catholic Church and the salvation of souls, called by Our Predecessor, Gregory XVI, an "insanity" [cf. his Mirari Vos],² viz., that "liberty of conscience and worship is each man's personal right, which ought to be legally proclaimed and asserted in every rightly constituted society [1]; and that a right resides in the citizens to an absolute liberty, which should be restrained by no authority whether ecclesiastical or civil, whereby they may be able openly and publicly [2] to manifest and declare any of their ideas whatever, either by word of mouth, by the press, or in any other way." But, while they rashly affirm this, they do not think and consider that they are preaching "liberty of perdition;"³ and that "if human arguments are always allowed free room for discussion, there will never be wanting men who will dare to resist truth, and to trust in the flowing speech of human wisdom; whereas we know, from the very teaching of our Lord Jesus Christ, how carefully Christian faith and wisdom should avoid this most injurious babbling."⁴

and to his Syllabus of Errors, which condemned the following:

15. Every man is free to embrace and profess that religion which, guided by the light of reason, he shall consider true. [1]—Allocution "Maxima quidem," June 9, 1862; Damnatio "Multiplices inter," June 10, 1851.

…

X. ERRORS HAVING REFERENCE TO MODERN LIBERALISM

77. In the present day it is no longer expedient that the Catholic religion should be held as the only religion of the State, to the exclusion of all other forms of worship.—Allocution "Nemo vestrum," July 26, 1855.

78. Hence it has been wisely decided by law, in some Catholic countries, that persons coming to reside therein shall enjoy the public exercise [2] of their own peculiar worship.—Allocution "Acerbissimum," Sept. 27, 1852.

79. Moreover, it is false that the civil liberty [1] of every form of worship, and the full power, given to all, of overtly and publicly manifesting any opinions whatsoever and thoughts, conduce more easily to corrupt the morals and minds of the people, and to propagate the pest of indifferentism.—Allocution "Nunquam fore," Dec. 15, 1856.

80. The Roman Pontiff can, and ought to, reconcile himself, and come to terms with progress, liberalism and modern civilization.—Allocution "Jamdudum cernimus," March 18, 1861.

^{(Taken from here, which is nicely color-coded; unfortunately, StackExchange markdown doesn't support colors.)}

Despite the SSPX asking for doctrinal clarity, the post-Vatican II magisterium has yet to clarify exactly how Dignitatis Humanæ is in continuity with 19^th century popes' magisterial statements. In fact, Cdl. Ratzinger said the Vatican II document Gaudium et Spes is a "counter-Syllabus" of Pius IX's Syllabus of Errors. Since the Church's ordinary and universal magisterium is infallible (i.e., cannot error), it follows that the Vatican II document on this issue is not magisterial.

A dangerous consequence of Dignitatis Humanæ was that States with constitutions that recognized the Catholic Church as the true religion (e.g., Spain) felt compelled to change their constitutions to view all religions equally.

^{cf. also these encyclicals condemning modern errors}