You can only have one unbound spirit summoned at a time
While the text on the book is vague about wether you can summon only one at once or control only one at once (core rulebook, page 300):
You can only summon one spirit at a time, and it only hangs around for a limited time
There is no mention about how many summoned spirits you can have at once, but there is about how many bound spirits you can have. Binding a spirit is much harder than summoning one, so why there would be a restriction on bound spirits but not on summoned ones?
Once the spirit is bound, then the spirit and its services
do not expire at the next sunrise or sunset. A spirit’s
service ends when it has no more services owed
to the magician. The bound spirit can be called or dismissed
with a Simple Action as they appear next to the
magician from the metaplane, awaiting further instructions
on the astral. A magician can bind up to his Charisma
attribute in spirits.
This is enforced by the Additional Purchases and Resitrictions table (page 98), which says:
1 Karma per service (Force of spirit is equal to Character’s Magic Attribute rating)
Can only have a number of bound spirits equal to Charisma
This was the text on 4th edition (core rulebook, page 188) and it was crystal clear on how they are supposed to work:
A magician may only have one unbound spirit summoned at any
given time, and no more bound spirits than her Charisma attribute.
Spirits on remote service and on standby count toward this total.
The term unbound spirit isn't even used on the 5th edition core rulebook. But refers to spirits not bound to your character and there is no reason to believe this has changed between editions, as what is a bound spirit has the same definition (see the Binding skill).
Historically, this ruling has been the same on the 3rd, 2nd and 1st edition of the game. Shamans could summon nature spirits, while mages could summon elementals. Both also had limits, but they were mechanically different.
Nature spirits couldnt be bound to your character, their services had to be used while they were summoned and they would vanished on sunrise or sunset (similar to summoning spirits in 4th and 5th edition). While elementals had to be bound in order to be used, which took some hours and money (similarl to binding spirits in 4th and 5th edition), but would not disapear until all services were consumed.
The limitations were similar to what we got on 5th edition:
A shaman can summon only one nature spirit or have one in service to him at one time.
A mage can bind, at once time, a number of elementals equal to his Charisma.
The significant differences were that nature spirits didn't require any preparation and could be used quickly, while elementals were stronger due to being on standby and materialize when needed, and had the option to follow more complex orders (Aid Sorcery, Aid Study, Remote Services, sustain spells, etc).
What actually happened on the 4th edition was that these rules were combined into a single subsystem that works for shamans and mages without major mechanical differences other than the type of spirits available and the flavor on how each tradition handles spirits.
Shadowrun Missions FAQ
This is also explicitly covered in the Missions FAQ (no official link yet, as their messageboards are offline, but here is an unofficial mirror):
How many Unbound Spirits can you have at one time? The text (SR5, pg. 300) isn’t very clear.
You may only have one unbound spirit at any given moment
As for the validity of the missions FAQ:
These changes are Errata for Missions only, and are subject to change when the official Errata gets released.
Among other mistakes, this is just one more example of the designers being lazy and not properly describing mechanics that hasn't changed between editions. The 5th edition core rulebook is full of information that were previously on splatbooks, but basic mechanics suffered, leading to a lot of errata removing leftover text from the 4ed core rulebook.