What does John Calvin mean to imply by the "suffrage" of the church?

Answer for question 1: Suffrage

Suffragio also means intercession.So Suffragio can mean suffrage , vote, intercession or a combination.

With this understanding of suffrage, vote and intercession. Lets take your posting and compare within a context that Calvin may be referring to.

"Thus, while the church receives and gives its seal of approval to the Scriptures, it does not thereby render authentic what is otherwise doubtful or controversial. But because the church recognizes Scripture to be the truth of its own God, as a pious duty it unhesitatingly venerates Scripture."

According to the above statement, let's look at a perfect example. One of the first ancient councils where "Suffragio", intercession and vote was needed.

Example:

The First Council of Nicea, was a case where The Church was under attack from heresies, Arians denying the divinity of Christ. Similar to the statement above declares. "it does not thereby render authentic what is otherwise doubtful or controversial"

Likewise the Council of Nicea did not render authentication of what was doubtful or controversial. It was the opposite, that is to say The Church was defending Apostolic teaching, defending what had been passed down and always understood and believed. Defending what they already knew. No new theology came from this council. Suffragio (intercession and vote) was needed because of outside heretical forces to defend the scripture. Thus the Nicene creed came into existence. Not creating anything new, but again defending what was already known, understood, believed and passed down form the Apostles. Everything being defending through suffrage,intercession and vote.

This example would also answer your question about a "communal expression of faith". The ancient Church believed that as long as the entire Church was united, and voted united, that the Holy Spirit was guiding them. This is very evident throughout the first seven ecumenical councils where "Suffragio" was required. In contrast, if any one part were to leave, there could no longer be ecumenical intercession or votes. A foot note that ever since the Great Schism, Rome being separated from the main body of the other four (Alexandria,Jerusalem,Antioch,Constantinople), there has never been another ecumenical council in the East, that included all. Rome continued with their own councils only effecting the West, for which eventually brought about Luther's and Calvin's writings..

You next inquiry is: "I'm asking to what extent "the Church" is present here as a single entity." This is a great question and a little complex depending on context. Most of Calvin's writings are against the Roman Church in some way or another. He may very well be making his statement in context of all other historical occurrences and examples before the Great Schism, in rebuttal to the Roman Church's changes in the West after the Schism. If he is writing in the context of Sola Scriptora, this would be a difficult position to defend, as other Protestant churches had already come into existence and there is no logical way he could pick from one of the many diverse entities who did not agree, claiming a single entity. The concept of an all encompassing invisible Church does not exist anywhere in Christian history before this. The Church is very visible and united for over a 1,000 years until the Great Schism.

Let's explore all possibilities he could be thinking. So "the Church" is present here as a single entity" would depend on context.

1. The original ancient Church that stood united for over 1,000 years is considered The Church.

2. After the Great Schism of 1054 AD, Rome claimed to be The Church.

3. Likewise The Church of the East, the Orthodox (Alexandria,Jerusalem,Antioch,Constantinople and Rome) claimed to be The Church and still claims this.

4. According to translation Calvin is using a present tense. "recognizes" which alludes to a more present understanding or what Calvin's understanding of "The Church".

   If the case is option 4, this would allude to an understanding only by Calvin. Basically his own understanding.

   Note: If the translation was in a past tense. "recognized", ex:'But because the church recognized Scripture to be the truth of its own God' I would say he would have been referring to past occurrences, when The Church was united. Maybe again as a rebuttal, to say Romes intercession's , votes were not in defense, but rather they were changing things. He still could be meaning this in one form or another.

Answer for question 2:Is there a meaningful concept of voting? In accordance to Church history, Yes! However, only when The Church (ALL) is fully united. Thus the reason there were never more than the seven ecumenical councils. Rome being separated in 1054AD, to this day, no other ecumenical council has ever taken place. Not because Rome was more important, Each bishop whether from Rome or Jerusalem was allowed only a vote a piece, it's more along the lines of The Church has been separated. I would suggest looking at the Ante-Nicene collection and the councils specifically to help you understand the need for the councils and voting in ancient times.

I would not think at all that Calvin purposely intends "aggregating everyone's opinions and picking the most popular". That would not match anything in Christian history. However he is in a little bit of a dilemma, writing against the Roman Church, but not being united with the Church from the East. All Churches will say the are being lead by the Holy Spirit, but don't agree.I personally think Calvin has a deeper understanding, however the position that he is in, the era, his surroundings, no contact with the East, is indeed a difficult one.

Answer for question 3: Is the parallel with Calvin's sacrament language just a coincidence, or are the same concepts being invoked in both instances? The sacramental language is not a coincidence. Again, based on history, the seven ecumenical councils of the East are considered Holy occurrences of Suffragio and defense within The Church. I'm very confident in my answer to say that Calvin, within his own understanding, is expressing a sacramental expression. I will simply express again, within his own understanding.

Due to lack of exact source reference where Calvin is said to use the word "suffrage", little can be said on that. Book I chapter 7 is mentioned, but no quote is given where Calvin speaks of "suffrage". Your first paragraph shows that Calvin did not think the Church ratified the biblical canon. Elsie Anne McKee is quoted, and she translated Calvin's French edition of 1541 (in 2009). McKee wrote that his French edition "was more than a translation of the Latin 1539; it was a text of pastoral theology in the language of the common people and consciously directed to them" (x). This suggests that 16th century French understanding of the word 'suffrage' must take precedence over modern ideas about that word. To 'seal' [ratify] and to 'vote' can give a misleading impression.

McKee shows that the Church received the scriptures, accepting them as the word of God. It is the seal of the Holy Spirit that confirms to Christians various things (Romans 8:9, 15-16 & 2 Timothy 2:19), including His inspiration of those holy scriptures. Calvin wrote that the external minister preaches and administers the sacraments, but "the internal minister, who is the Holy Spirit... effects in the hearts of whomsoever he will their union with Christ." (Calvin, Theological Treatises, 173)

The best part of the Institutes to go to for Calvin's views on the Church come in Book IV (a third of the 1559 Institutes). The four Books go up in a spiral: Book I, Knowledge of God the Creator, would do us no good (Calvin said) without Book II - Knowledge of God our Redeemer in Christ. But that does not benefit us unless we are united to him by the Holy Spirit (the topic of Book III). And it is by the ministry of the Church (the topic of Book IV) that God begets and nourishes his children. McKee writes:

"If the heart of piety, 'reverence and love of God', is clearly the work of the Holy Spirit, the ordained earthly instrument for building up that piety is the church" John Calvin, Writings on Pastoral Piety, p. 23

Once Calvin's emphasis on the role of the Holy Spirit in the Church is seen, then we can understand his concept of the Church. In Book IV he sets forth "the external means" God uses to bring us to himself and to preserve us in the faith: "the church, its government, orders, and power; then the sacraments; and lastly, the civil order." (IV.1.1)

You ask "to what extent 'the Church' is present here as a single entity." There is only, ever, one Church, comprised of many members (as Paul explained). See Calvin's Sermon on Ephesians chapters 3 and 4. Here is a quote from a book designed to help us read Calvin's Institutes devotionally:

"Preaching, the sacraments, and church discipline are the means by which God 'holds us therein'. God does not raise us to perfection in a moment but makes us grow 'little by little under the nurture of the church' (IV.1.5) The place of our Christian growth is not in our isolated individual lives, but among the congregation of believers, where 'all the blessings which God bestows upon them are mutually communicated to each other' (IV.1.3).

...Calvin makes it clear that 'although God's power is not bound to outward means, he has nonetheless bound us to this ordinary manner of teaching' (TV.1.5). In other words, God could do without the church, but we cannot." Knowing God And Ourselves, David B. Calhoun, p 260, The Banner of Truth Trust, 2016

With regard to your second question: "Is there a meaningful concept of voting?" - This is not about "the approval of the Church". Nor is it about today's modern enthusiasm for individuality and independence. Quite the opposite. The Church, as a collective of God's people, depends on the approval of God working through the unifying effect of the Holy Spirit in 'the visible Church'. The modern idea of individuals who have their names on a congregation's membership list voting to approve decisions, so that a majority decision is accepted, is not Calvin's idea of 'the suffrage of the Church'. Yes, elders are to agree on decisions, but their deliberations are to be based on the authority of God's word and the leading of the Holy Spirit. Everything in the Church has to be subservient to that. Calvin knew that schismatics, even some wicked people, might be present in a congregation but that "In estimating the true church, divine judgment is of more weight than human" (IV.1.16). The system of Church governance and decision-making followed that of the first century Church, as shown in the New Testament, carried out by elders and deacons.