For fungi, you can use ITS1 and ITS2 regions in the same way as 16S rRNA gene in case of bacteria.
Once, you have ITS regions (sequences) you can use BLAST or RDP Classifier from QIIME using UNITE database as reference. Here in case of BLAST, you can specifiy sequence similarity cutoffs. like, >97% means species level identification.
Use can also use KAIJU webserver, where you can upload combined/stitched ITS reads and can choose "NCBI BLAST nr +euk" (It includes eukaryotes along with fungi and microbial eukaryotes) as reference database to consider. It is sensitive and still very fast. Alternatively you can download and install its standalone version.