Is 'bhfuil' a verb?

Some of the examples I've seen, for example, "An bhfuil Cait anseo?" seem to treat "Bhfuil" or some construction based on it (an bhfuil/nach bhfuil/go bhfuil) as a verb meaning roughly 'to be'. Others are translated into English where we would use do-insertion in terms of syntax, e.g.

"An bhfuil Gaelige agat?"/"Do you speak Irish?/

Does bhfuil mean something like "to have" i mBáerla?

Could someone help?

Thanks

A handful of irregular verbs have separate dependent and independent forms. When they do, the independent form is used, well, independently, and the dependent form with verbal particles. Depending on the particle, the dependent form may also be lenited or eclipsed.

That's what you're seeing here. The basic verb is "tá" (= 'to be'), and its dependent form is "fuil."
Tá sé anseo. - He is here.
Níl (orig. 'ní fhuil') sé anseo. - He is not here.
An bhfuil sé anseo? - Is he here?
Nach bhfuil sé anseo? - Isn't he here?

And there isn't really a verb 'to have' in Irish, so you normally express possession with 'tá' plus the preposition 'ag':
Tá deirfiúr ag Máirtín. - Máirtín has a sister.
An bhfuil deirfiúr ag Máirtín? - Does Máirtín have a sister?
etc.

And to add that bit about languages:

In Irish you say that there is a language at you or to use the interpretation that Abigail gave that you have a language, a form you can also find in the English spoken in Ireland. Thus it is "Tá Gaeilge agam." - I have Irish.

In Korean the verb 'to be' does the job of signally ownership, tho less perphrastically than Irish. It is not a surprise to find students say 'I'm book', thus...

A Bhearn,

Not sure what you mean here. There are two constructions that spring to mind: (1) The indirect with the verb of existence 있다 /iss.ta/ and a dative postposition -에게/-께/-한테 /-eykey/, /-kkey/, /-hanthey/ indicating the possesser (e.g. 나한테 책이 있다 /na hanthey chayk.i iss.ta/), which is exactly as periphrastic as the Irish equivalent, and (2) the direct with the verb of possession 가지다 /kacita/ (e.g. 내가 책을 가진다 /nay ka chayk ul kacinta/), which is no more periphrastic than English "I have a book".

I take back what I said about being periphrastic, then

Irish utilizes prepositional pronouns, which are contractions of a base preposition and a pronoun, in some rather confusing ways. This is one of the many influences behind Hiberno-English.

I am hungry translates as Tá ocras orm which literally means There is hunger on me. Orm being a contraction of Ar meaning On and mé

An bhfuil Gaelige agat? translates as Do you speak Irish but literally in means Is Irish at you? which is to say Do you possess any Irish? which leads to the Hiberno-English Do you have any Irish?

Once you can get your mind wrapped around the prepositional pronouns understanding possession or possessive relationships gets a bit easier which isn't to say that it's exactly easy. The hard part is learning which preposition to use with which phrases. Is it agam or orm? Is it chugat or duit? The more you're exposed to the usage, the more it you'll be able to discern what sounds right.

I hope this doesn't overwhelm you but more importantly, I hope this was useful.