No, familiars can be gained by means other than summoning
So there are three broad ways you can gain a familiar:
By summoning one, using the find familiar spell;
By forming a pact with a creature who will willingly be your familiar;
By creating one yourself
In the first case, you — the player — have control over the actions of the familiar, as detailed in the spell description. But for the second case, the DM controls the familiar's actions, not you.
In other words, if you find a real pseudodragon in the game, tame it, and convince it to serve you, it becomes your familiar, but it is still an NPC. At any time, the DM may decide that you have violated the terms of your agreement/broken your mutual trust, and the familiar can leave you or betray you. It is choosing to be with you, but you don't get to decide its motivations.
It is also possible to have more than one familiar this way, if you summon one and form a pact with another one.
The Imp: the familiar that can be forged a pact with
In MM pg 69, you will find on the sidebar an "Imp Familiar" variant. They are described as:
Imps can be found in the service to mortal spellcasters, acting as advisors, spies, and familiars. An imp urges its master to acts of evil, knowing the mortal's soul is a prize the imp might ultimately claim. […]
Familiar. The imp can enter into a contract to serve another creature as a familiar, forming a telepathic bond with its willing master. […]
So there is the second way to get a familiar, as demonstrated in the RAW.
The Homunculus: the familiar that can be created
In MM pg 188, you will see the Homunculus: a tiny creature that is made by the wizard — not summoned or made a pact with. The text does not actually say the word "familiar," but its abilities as a companion are everything a familiar can do, and a little extra.
A homunculus is a construct that acts as an extension of its creator, with the two sharing thoughts, senses, and language through a mystical bond. A master can have only one homunculus at a time (attempts to create another one always fail), and when its master dies, the homunculus also dies.
Shared Mind. A homunculus knows everything its creator knows, including all the languages the creator can speak and read. Likewise, everything the construct senses is known to its master, even over great distances, provided both are on the same plane.
The implications are minor, but not totally zero. To address your specific questions:
The type of spirit does not cause any visible differences, unless the player or the DM want it to. That would be a flavor/reskinning call, not a rules or mechanics call. Note that the spirit type and the shape taken are not linked or restricted in any way. A stark example is if you are a Pact of the Chain Warlock, you familiar can take the shape of an Imp, Quasit, Sprite, or Pseudodragon, and this is given no limitation as to the type of spirit. So you could have a Celestial Quasit.
The type may have an effect on the spirit's alignment, but again, that's up to the player and the DM. Again, that would be a flavor/reskinning call, not a rules or mechanics call. It obeys your wishes and commands regardless. It would be up to him to decide if his Celestial Quasit is mannerly, polite, and never uses profanity.
Yes, the default is that the character gets the same spirit back, just in a new shape. The caster does have the ability to dismiss a familiar forever, so presumably if he does so, casting it again will get him a different spirit, and could choose a different type. Again, flavor/reskinning issues, up to the player and the DM. Certainly, if the DM allows him to change the spirit, getting one of a different type would mean a new spirit, and thus a new NPC, with no memories of what the last one experienced.
The main actual effects, as far as the rules go, are that the type of the creature is celestial, fey, or fiend, (and not beast), and this may affect what spells it is affected by (e.g. immune to Animal Friendship, but may be subject to Magic Circle. Also, depending on its type, a Paladin's Divine Sense may detect it, and that could tell them something about the character (or at least who he associates with).
The point of it being a spirit, from a mechanics standpoint, is mostly to explain where it comes from, why it is obedient and maybe more intelligent than normal, and to provide a mechanism for summoning, re-summoning, and dismissal.
How much control the character has over the form of the familiar is, again, up to the player and DM. If I, Gimble the Great, decide I want an owl, is that all I can specify, and the spirit decides if it will be a 4-ounce burrowing owl, or a 5-pound snowy owl, and all the details of its appearance? Or could I specify the species of owl, but it gets to decide the coloration? Or do I get to pick all the details myself?
Best Answer
From Find Familiar (PHB p. 294)
They have the same Intelligence (and Wisdom, and Strength etc.) as the chosen form.
The spell is explicit about what they can and cannot do. Specifically:
Of these the only one where lack of intelligence would be an impediment would be obeying commands. The forms mentioned all have an Intelligence of 1, 2 or 3, so: